Von: superseq-reply@research.att.com
Gesendet: 11 March, 2004 08:40
An: Hugo.Pfoertner@muc.mtu.de
Betreff: Reply from superseeker
Report on [ 5,21,85,341,1365,5461,21845,87381]:
Many tests are carried out, but only potentially useful information
(if any) is reported here.
TEST: IS THE SEQUENCE OF ABSOLUTE VALUES IN THE ENCYCLOPEDIA?
Matches (up to a limit of 50) found for 5 21 85 341 1365 5461 21845 87381 :
%I A002450 M3914 N1608
%S A002450 0,1,5,21,85,341,1365,5461,21845,87381,349525,1398101,5592405,22369621,
%T A002450 89478485,357913941,1431655765,5726623061,22906492245,91625968981,
%U A002450 366503875925,1466015503701,5864062014805,23456248059221,93824992236885
%N A002450 (4^n - 1)/3.
%C A002450 For n>0, a(n) is the degree (n-1) "numbral" power of 5 (see A048888 for the definition of numbral arithmetic). Example: a(3)=21, since the numbral square of 5 is 5(*)5 = 101(*)101(base 2) = 101 OR 10100 = 10101(base 2) = 21, where the OR is taken bitwise. - John W. Layman (layman(AT)math.vt.edu), Dec 18 2001
%C A002450 a(n) is composite for all n > 2 and has factors x, (3x+2(-1)^n) where x belongs to A001045. In binary the terms are 1, 101, 10101, 1010101, etc. - John McNamara (mistermac39(AT)yahoo.com), Jan 16 2002
%C A002450 Number of n X 2 binary arrays with path of adjacent 1's from upper left corner to right column. - Ron Hardin (rhh(AT)cadence.com), Mar 16 2002
%C A002450 Collatz-function iteration started at with a[n] will surely ended by 1 in exactly 2n steps. - Labos E. (labos(AT)ana1.sote.hu), Sep 30 2002
%C A002450 Also sum of squares of divisors of 2^n: a(n)=A001157[A000079(n)] - Paul Barry (pbarry(AT)wit.ie), Apr 11 2003
%C A002450 All members of sequence are also generalized octagonal numbers (A001082). - Matthew Vandermast (ghodges14(AT)msn.com), Apr 10 2003
%C A002450 Binomial transform of A000244 (with leading zero) - Paul Barry (pbarry(AT)wit.ie), Apr 11 2003
%D A002450 A. Fletcher, J. C. P. Miller, L. Rosenhead, and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 112.
%D A002450 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
%D A002450 T. N. Thiele, Interpolationsrechnung. Teubner, Leipzig, 1909, p. 35.
%H A002450 H. Bottomley, Illustration of initial terms
%H A002450 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 373
%H A002450 E. W. Weisstein, Link to a section of The World of Mathematics.
%F A002450 a(n+1)= sum(A060921(n,m),m=0..n). G.f.: x/(1-5*x+4*x^2). - Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Apr 24 2001
%F A002450 a(n)= sum{k=0..n-1, 4^k} a(n)= A001045(2n). - Paul Barry (pbarry(AT)wit.ie), Mar 17 2003
%F A002450 Second binomial transform of A001045. - Paul Barry (pbarry(AT)wit.ie), Mar 28 2003
%F A002450 E.g.f. (exp(4x)-exp(x))/3 - Paul Barry (pbarry(AT)wit.ie), Mar 28 2003
%F A002450 a(0) = 0, a(n+1) = 4*a(n) + 1 . - DELEHAM Philippe (kolotoko(AT)lagoon.nc), Feb 25 2004
%p A002450 [seq((4^n-1)/3,n=0..40)];
%Y A002450 a(n) = (A007583(n)-1)/2.
%Y A002450 Partial sums of powers of 4, A000302.
%Y A002450 a(n)=A000975(2n)/2.
%Y A002450 A084160(n) = 2*a(n).
%Y A002450 Cf. A002446, A024036, A084180, A080674, A047849.
%Y A002450 Sequence in context: A026027 A002054 A028948 this_sequence A084241 A026855 A012814
%Y A002450 Adjacent sequences: A002447 A002448 A002449 this_sequence A002451 A002452 A002453
%K A002450 nonn,easy,nice
%O A002450 0,3
%A A002450 njas
%I A084241
%S A084241 0,1,5,21,85,341,1365,5461,21845,87381,349525,1398101,5592405,22369621,
%T A084241 89478485,357913941,1431655765,5726623061,22906492245,91625968981,
%U A084241 366503875925,1466015503701,5864062014805,23456248059221,93824992236885
%V A084241 0,1,-5,21,-85,341,-1365,5461,-21845,87381,-349525,1398101,-5592405,22369621,-89478485,
%W A084241 357913941,-1431655765,5726623061,-22906492245,91625968981,-366503875925,1466015503701,
%X A084241 -5864062014805,23456248059221,-93824992236885
%N A084241 a(n)=-5a(n-1)-4a(n-2), a(0)=0,a(1)=1.
%C A084241 abs(a(n))=A002450(n)=A001045(2n). Binomial transform is (0,1,-3,9,-27,...).
%F A084241 a(n)=((-1)^n-(-4)^n)/3; a(n)=sum{k=1..n, (-1)^(n+k)binomial(n,k)(-3)^(k-1) }; G.f.: x/((1+x)(1+4x)); E.g.f.: (exp(-x)-exp(-4x))/3.
%Y A084241 Cf. A084240.
%Y A084241 Sequence in context: A002054 A028948 A002450 this_sequence A026855 A012814 A039919
%Y A084241 Adjacent sequences: A084238 A084239 A084240 this_sequence A084242 A084243 A084244
%K A084241 easy,sign
%O A084241 0,3
%A A084241 Paul Barry (pbarry(AT)wit.ie), May 21 2003
SUCCESS: the sequence is in the table.
SUGGESTION: GUESSGF FOUND ONE OR MORE GENERATING FUNCTIONS
WARNING: THESE MAY BE ONLY APPROXIMATIONS!
Generating function(s) and type(s) are:
5 - 4 x
[- ---------------, ogf]
2
-4 x - 1 + 5 x
[- 1/3 exp(x) + 16/3 exp(4 x), egf]
[- 1/3 exp(x) + 16/3 exp(4 x), egf]
SUGGESTION: LISTTOALGEQ FOUND ONE OR MORE ALGEBRAIC
EQUATIONS SATISFIED BY THE GEN. FN.
WARNING: THESE MAY BE ONLY APPROXIMATIONS!
Equation(s) and type(s) are:
2
[n + (-5 - 5 n) a(n) + (4 + 4 n) a(n) , revogf]
Types of generating functions that may have been mentioned above:
ogf = ordinary generating function
egf = exponential generating function
revogf = reversion of ordinary generating function
revegf = reversion of exponential generating function
lgdogf = logarithmic derivative of ordinary generating function
lgdegf = logarithmic derivative of exponential generating function
TRY "GUESSS", HARM DERKSEN'S PROGRAM FOR GUESSING A GENERATING FUNCTION FOR A SEQUENCE.
Guesss - guess a sequence, by Harm Derksen (hderksen@math.mit.edu)
Guesss suggests that the generating function F(x)
may satisfy the following algebraic or differential equation:
x-5/4+(x^2-5/4*x+1/4)*F(x) = 0
If this is correct the next 6 numbers in the sequence are:
[349525, 1398101, 5592405, 22369621, 89478485, 357913941]
TEST: APPLY VARIOUS TRANSFORMATIONS TO SEQUENCE AND LOOK IT
UP IN THE ENCYCLOPEDIA AGAIN
SUCCESS
(limited to 40 matches):
Transformation T011 gave a match with:
%I A020988
%S A020988 2,10,42,170,682,2730,10922,43690,174762,699050,2796202,
%T A020988 11184810,44739242,178956970,715827882,2863311530,11453246122,
%U A020988 45812984490,183251937962,733007751850,2932031007402,11728124029610
%N A020988 2(2^{2n+2} - 1)/3.
%C A020988 Expected time to finish a random Tower of Hanoi problem with 2n+2 disks using optimal moves, so (since 2n+2 is even and A010684(2n+2)=1) a(n)=A060590(2n+2). - Henry Bottomley (se16(AT)btinternet.com), Apr 05 2001
%C A020988 a(n)=number of derangements of [2n+3] with runs consisting of consecutive integers. E.g. a(1)=10 because the derangements of {1,2,3,4,5} with runs consisting of consecutive integers are 5|1234, 45|123, 345|12, 2345|1, 5|4|123, 5|34|12, 45|23|1, 345|2|1, 5|4|23|1, 5|34|2|1 (the bars delimit the runs). - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 26 2003
%C A020988 a(n) = A007583(n+1)-1 = A039301(n+2)-2 = A083584(n)+1 = A084180(n+1). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Jun 14 2003
%C A020988 For n>0 also smallest numbers having in binary representation exactly n+1 maximal groups of consecutive zeros: A087120(n)=a(n-1), see A087116. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Aug 14 2003
%D A020988 J. Brillhart and P. Morton, A case study in mathematical research: the Golay-Rudin-Shapiro sequence, Amer. Math. Monthly, 103 (1996) 854-869.
%F A020988 a(n)=4*a(n-1) + 2 - Lee Hae-hwang (mathmaniac(AT)empal.com), Sep 24 2002
%Y A020988 A020988(n) = A026644(2n)
%Y A020988 3 * A014131(n) = A026644(n) 2 * A000975(n) = A026644(n) 2 + A052953(n) = A026644(n) 1 + A001045(n) = A026644(n).
%Y A020988 Sequence in context: A083937 A085224 A024483 this_sequence A084180 A084480 A005144
%Y A020988 Adjacent sequences: A020985 A020986 A020987 this_sequence A020989 A020990 A020991
%K A020988 nonn
%O A020988 0,1
%A A020988 njas
%I A084180
%S A084180 0,2,10,42,170,682,2730,10922,43690,174762,699050,2796202,11184810,
%T A084180 44739242,178956970,715827882,2863311530,11453246122,45812984490,
%U A084180 183251937962,733007751850,2932031007402,11728124029610,46912496118442
%N A084180 a(n)=4a(n-1)+2, a(0)=0.
%C A084180 Row sums of triangle using cumulative sums of odd-indexed rows of Pascal's triangle (start with zeros for completeness) :
%C A084180 . . . . 0 . 0 . . . .
%C A084180 . . . . 1 . 1 . . . .
%C A084180 . . . 1 4 . 4 1 . . .
%C A084180 . . 1 6 14 14 6 1 . .
%C A084180 .1 8 27 49 49 27 8 1.
%C A084180 a(n) = A007583(n)-1 = A020988(n-1) = A039301(n+1)-2 = A083584(n-1)+1. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Jun 14 2003
%F A084180 a(n)=(2/3)(4^n-1); G.f. : 2x/((1-x)(1-4x)); E.g.f. : (2/3)(exp(4x)-exp(x)).
%Y A084180 a(n)/2=A002450(n).
%Y A084180 Cf. A020989.
%Y A084180 Sequence in context: A085224 A024483 A020988 this_sequence A084480 A005144 A064835
%Y A084180 Adjacent sequences: A084177 A084178 A084179 this_sequence A084181 A084182 A084183
%K A084180 easy,nonn
%O A084180 0,2
%A A084180 Paul Barry (pbarry(AT)wit.ie), May 18 2003
%I A087120
%S A087120 1,0,10,42,170,682,2730,10922,43690,174762,699050,2796202,11184810,
%T A087120 44739242,178956970,715827882,2863311530,11453246122,45812984490,
%U A087120 183251937962,733007751850,2932031007402,11728124029610,46912496118442
%N A087120 Smallest numbers having in binary representation exactly n maximal groups of consecutive zeros.
%C A087120 A087116(a(n))=n and A087116(k)1: a(n) = A020988(n-1).
%H A087120 Inde x entries for sequences related to binary expansion of n
%F A087120 a(0)=1, a(1)=0, a(2)=10, a(n)=4*a(n-1)+2.
%Y A087120 Cf. A087118, A023416, A007088.
%Y A087120 Sequence in context: A031146 A027171 A003822 this_sequence A007226 A027149 A077541
%Y A087120 Adjacent sequences: A087117 A087118 A087119 this_sequence A087121 A087122 A087123
%K A087120 nonn
%O A087120 0,3
%A A087120 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Aug 14 2003
Transformation T020 gave a match with:
%I A055841
%S A055841 1,2,9,36,144,576,2304,9216,36864,147456,589824,2359296,9437184,
%T A055841 37748736,150994944,603979776,2415919104,9663676416,38654705664,
%U A055841 154618822656,618475290624,2473901162496
%N A055841 A second order recursive sequence.
%C A055841 First differences of A002001.
%D A055841 A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pps. 194-196.
%F A055841 a(n)=9*4^(n-2), a(0)=1, a(1)=2.
%e A055841 a(n)=4a(n-1)+[(-1)^n]*C(2,2-n). G.f.(x)=(1-x)^2/(1-4x).
%Y A055841 Cf. A000302 and A002001.
%Y A055841 Sequence in context: A027995 A077836 A003125 this_sequence A037521 A037730 A029874
%Y A055841 Adjacent sequences: A055838 A055839 A055840 this_sequence A055842 A055843 A055844
%K A055841 easy,nonn
%O A055841 0,2
%A A055841 Barry E. Williams, May 30 2000
%I A002063
%S A002063 9,36,144,576,2304,9216,36864,147456,589824,2359296,9437184,
%T A002063 37748736,150994944,603979776,2415919104,9663676416,38654705664,
%U A002063 154618822656,618475290624,2473901162496,9895604649984
%N A002063 9*4^n.
%Y A002063 Sequence in context: A000537 A023872 A034557 this_sequence A075674 A038780 A073984
%Y A002063 Adjacent sequences: A002060 A002061 A002062 this_sequence A002064 A002065 A002066
%K A002063 nonn
%O A002063 0,1
%A A002063 njas
Transformation T012 gave a match with:
%I A024036
%S A024036 0,3,15,63,255,1023,4095,16383,65535,262143,1048575,4194303,
%T A024036 16777215,67108863,268435455,1073741823,4294967295,17179869183,
%U A024036 68719476735,274877906943,1099511627775
%N A024036 4^n-1.
%D A024036 G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
%Y A024036 Equals 3 * A002450(n).
%Y A024036 Sequence in context: A072336 A067562 A062211 this_sequence A083858 A080948 A001447
%Y A024036 Adjacent sequences: A024033 A024034 A024035 this_sequence A024037 A024038 A024039
%K A024036 nonn
%O A024036 0,2
%A A024036 njas
Transformation T018 gave a match with:
%I A075877
%S A075877 1,2,3,4,5,6,7,8,9,1,1,1,1,1,1,1,1,1,1,1,2,4,8,16,32,64,128,256,512,1,
%T A075877 3,9,27,81,243,729,2187,6561,19683,1,4,16,64,256,1024,4096,16384,65536,
%U A075877 262144,1,5,25,125,625,3125,15625,78125,390625,1953125,1,6,36,216,1296
%N A075877 Powering the decimal digits of n.
%C A075877 a(n)=1 iff the initial digit is 1 or n contains a 0 (i.e. A055641(n)>0 or A000030(n)=1);
%C A075877 a(A011540(n))=1.
%F A075877 a(n) = a(floor(n\10))^(n mod 10).
%e A075877 a(253) = 2^5^3 = 32^3 = 32768.
%Y A075877 Cf. A007953, A007954.
%Y A075877 Sequence in context: A076313 A055017 A040997 this_sequence A052423 A000030 A004427
%Y A075877 Adjacent sequences: A075874 A075875 A075876 this_sequence A075878 A075879 A075880
%K A075877 nonn,base
%O A075877 1,2
%A A075877 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Oct 16 2002
%I A000302 M3518 N1428
%S A000302 1,4,16,64,256,1024,4096,16384,65536,262144,1048576,4194304,16777216,
%T A000302 67108864,268435456,1073741824,4294967296,17179869184,68719476736,274877906944,
%U A000302 1099511627776,4398046511104,17592186044416,70368744177664,281474976710656
%N A000302 Powers of 4.
%C A000302 Same as Pisot sequences E(1,4), L(1,4), P(1,4), T(1,4). See A008776 for definitions of Pisot sequences.
%C A000302 The convolution square root of this sequence is A000984, the central binomial coefficients: C(2n,n). - T. D. Noe (noe(AT)sspectra.com), Jun 11 2002
%C A000302 a(n)=sum(k=0,n,C(2k,k)*C(2(n-k),n-k)). - Benoit Cloitre (abcloitre(AT)wanadoo.fr), Jan 26 2003
%H A000302 Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%H A000302 P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000302 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 8
%H A000302 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 269
%H A000302 Index entries for "core" sequences
%F A000302 a(n) = 4^n; a(n) = 4a(n-1).
%F A000302 G.f.: 1/(1-4x), e.g.f.: exp(4x)
%F A000302 1 = Sum(n = 1 through infinity) 3/a(n) = 3/4 + 3/16 + 3/64 + 3/256 + 3/1024...; with partial sums: 3/4, 15/16, 63/64, 255/256, 1023/1024... - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 16 2003
%p A000302 A000302:=n->4^n;
%Y A000302 Cf. A024036, A052539.
%Y A000302 Sequence in context: A006811 A005755 A077821 this_sequence A050734 A075614 A083592
%Y A000302 Adjacent sequences: A000299 A000300 A000301 this_sequence A000303 A000304 A000305
%K A000302 easy,nonn,nice,core
%O A000302 0,2
%A A000302 njas
Transformation T004 gave a match with:
%I A002450 M3914 N1608
%S A002450 0,1,5,21,85,341,1365,5461,21845,87381,349525,1398101,5592405,22369621,
%T A002450 89478485,357913941,1431655765,5726623061,22906492245,91625968981,
%U A002450 366503875925,1466015503701,5864062014805,23456248059221,93824992236885
%N A002450 (4^n - 1)/3.
%C A002450 For n>0, a(n) is the degree (n-1) "numbral" power of 5 (see A048888 for the definition of numbral arithmetic). Example: a(3)=21, since the numbral square of 5 is 5(*)5 = 101(*)101(base 2) = 101 OR 10100 = 10101(base 2) = 21, where the OR is taken bitwise. - John W. Layman (layman(AT)math.vt.edu), Dec 18 2001
%C A002450 a(n) is composite for all n > 2 and has factors x, (3x+2(-1)^n) where x belongs to A001045. In binary the terms are 1, 101, 10101, 1010101, etc. - John McNamara (mistermac39(AT)yahoo.com), Jan 16 2002
%C A002450 Number of n X 2 binary arrays with path of adjacent 1's from upper left corner to right column. - Ron Hardin (rhh(AT)cadence.com), Mar 16 2002
%C A002450 Collatz-function iteration started at with a[n] will surely ended by 1 in exactly 2n steps. - Labos E. (labos(AT)ana1.sote.hu), Sep 30 2002
%C A002450 Also sum of squares of divisors of 2^n: a(n)=A001157[A000079(n)] - Paul Barry (pbarry(AT)wit.ie), Apr 11 2003
%C A002450 All members of sequence are also generalized octagonal numbers (A001082). - Matthew Vandermast (ghodges14(AT)msn.com), Apr 10 2003
%C A002450 Binomial transform of A000244 (with leading zero) - Paul Barry (pbarry(AT)wit.ie), Apr 11 2003
%D A002450 A. Fletcher, J. C. P. Miller, L. Rosenhead, and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 112.
%D A002450 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
%D A002450 T. N. Thiele, Interpolationsrechnung. Teubner, Leipzig, 1909, p. 35.
%H A002450 H. Bottomley, Illustration of initial terms
%H A002450 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 373
%H A002450 E. W. Weisstein, Link to a section of The World of Mathematics.
%F A002450 a(n+1)= sum(A060921(n,m),m=0..n). G.f.: x/(1-5*x+4*x^2). - Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Apr 24 2001
%F A002450 a(n)= sum{k=0..n-1, 4^k} a(n)= A001045(2n). - Paul Barry (pbarry(AT)wit.ie), Mar 17 2003
%F A002450 Second binomial transform of A001045. - Paul Barry (pbarry(AT)wit.ie), Mar 28 2003
%F A002450 E.g.f. (exp(4x)-exp(x))/3 - Paul Barry (pbarry(AT)wit.ie), Mar 28 2003
%F A002450 a(0) = 0, a(n+1) = 4*a(n) + 1 . - DELEHAM Philippe (kolotoko(AT)lagoon.nc), Feb 25 2004
%p A002450 [seq((4^n-1)/3,n=0..40)];
%Y A002450 a(n) = (A007583(n)-1)/2.
%Y A002450 Partial sums of powers of 4, A000302.
%Y A002450 a(n)=A000975(2n)/2.
%Y A002450 A084160(n) = 2*a(n).
%Y A002450 Cf. A002446, A024036, A084180, A080674, A047849.
%Y A002450 Sequence in context: A026027 A002054 A028948 this_sequence A084241 A026855 A012814
%Y A002450 Adjacent sequences: A002447 A002448 A002449 this_sequence A002451 A002452 A002453
%K A002450 nonn,easy,nice
%O A002450 0,3
%A A002450 njas
%I A084241
%S A084241 0,1,5,21,85,341,1365,5461,21845,87381,349525,1398101,5592405,22369621,
%T A084241 89478485,357913941,1431655765,5726623061,22906492245,91625968981,
%U A084241 366503875925,1466015503701,5864062014805,23456248059221,93824992236885
%V A084241 0,1,-5,21,-85,341,-1365,5461,-21845,87381,-349525,1398101,-5592405,22369621,-89478485,
%W A084241 357913941,-1431655765,5726623061,-22906492245,91625968981,-366503875925,1466015503701,
%X A084241 -5864062014805,23456248059221,-93824992236885
%N A084241 a(n)=-5a(n-1)-4a(n-2), a(0)=0,a(1)=1.
%C A084241 abs(a(n))=A002450(n)=A001045(2n). Binomial transform is (0,1,-3,9,-27,...).
%F A084241 a(n)=((-1)^n-(-4)^n)/3; a(n)=sum{k=1..n, (-1)^(n+k)binomial(n,k)(-3)^(k-1) }; G.f.: x/((1+x)(1+4x)); E.g.f.: (exp(-x)-exp(-4x))/3.
%Y A084241 Cf. A084240.
%Y A084241 Sequence in context: A002054 A028948 A002450 this_sequence A026855 A012814 A039919
%Y A084241 Adjacent sequences: A084238 A084239 A084240 this_sequence A084242 A084243 A084244
%K A084241 easy,sign
%O A084241 0,3
%A A084241 Paul Barry (pbarry(AT)wit.ie), May 21 2003
Transformation T024 gave a match with:
%I A020989
%S A020989 1,6,26,106,426,1706,6826,27306,109226,436906,1747626,6990506,
%T A020989 27962026,111848106,447392426,1789569706,7158278826,28633115306,
%U A020989 114532461226,458129844906,1832519379626,7330077518506
%N A020989 (5*4^n-2)/3.
%C A020989 Let Zb[n](x) = polynomial in x whose coefficients are the corresponding digits of index n in base b. Then Z2[(5*4^k-2)/3](1/tau) = 1 - Marc LeBrun (mlb(AT)well.com), Mar 01 2001
%C A020989 a(n)=number of derangements of [2n+2] with runs consisting of consecutive integers. E.g. a(1)=6 because the derangements of {1,2,3,4} with runs consisting of consecutive integers are 4|123, 34|12, 4|3|12, 4|3|2|1, 234|1, and 34|2|1 (the bars delimit the runs). - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 26 2003
%D A020989 J. Brillhart and P. Morton, A case study in mathematical research: the Golay-Rudin-Shapiro sequence, Amer. Math. Monthly, 103 (1996) 854-869.
%F A020989 a(0)=1, a(n) = 4*a(n-1) + 2; a(n) = a(n-1)+ 5*{4^(n-1)}; - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), May 27 2001
%Y A020989 Sequence in context: A034560 A037545 A027996 this_sequence A079675 A046647 A046233
%Y A020989 Adjacent sequences: A020986 A020987 A020988 this_sequence A020990 A020991 A020992
%K A020989 nonn
%O A020989 0,2
%A A020989 njas
Transformation T005 gave a match with:
%I A002450 M3914 N1608
%S A002450 0,1,5,21,85,341,1365,5461,21845,87381,349525,1398101,5592405,22369621,
%T A002450 89478485,357913941,1431655765,5726623061,22906492245,91625968981,
%U A002450 366503875925,1466015503701,5864062014805,23456248059221,93824992236885
%N A002450 (4^n - 1)/3.
%C A002450 For n>0, a(n) is the degree (n-1) "numbral" power of 5 (see A048888 for the definition of numbral arithmetic). Example: a(3)=21, since the numbral square of 5 is 5(*)5 = 101(*)101(base 2) = 101 OR 10100 = 10101(base 2) = 21, where the OR is taken bitwise. - John W. Layman (layman(AT)math.vt.edu), Dec 18 2001
%C A002450 a(n) is composite for all n > 2 and has factors x, (3x+2(-1)^n) where x belongs to A001045. In binary the terms are 1, 101, 10101, 1010101, etc. - John McNamara (mistermac39(AT)yahoo.com), Jan 16 2002
%C A002450 Number of n X 2 binary arrays with path of adjacent 1's from upper left corner to right column. - Ron Hardin (rhh(AT)cadence.com), Mar 16 2002
%C A002450 Collatz-function iteration started at with a[n] will surely ended by 1 in exactly 2n steps. - Labos E. (labos(AT)ana1.sote.hu), Sep 30 2002
%C A002450 Also sum of squares of divisors of 2^n: a(n)=A001157[A000079(n)] - Paul Barry (pbarry(AT)wit.ie), Apr 11 2003
%C A002450 All members of sequence are also generalized octagonal numbers (A001082). - Matthew Vandermast (ghodges14(AT)msn.com), Apr 10 2003
%C A002450 Binomial transform of A000244 (with leading zero) - Paul Barry (pbarry(AT)wit.ie), Apr 11 2003
%D A002450 A. Fletcher, J. C. P. Miller, L. Rosenhead, and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 112.
%D A002450 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
%D A002450 T. N. Thiele, Interpolationsrechnung. Teubner, Leipzig, 1909, p. 35.
%H A002450 H. Bottomley, Illustration of initial terms
%H A002450 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 373
%H A002450 E. W. Weisstein, Link to a section of The World of Mathematics.
%F A002450 a(n+1)= sum(A060921(n,m),m=0..n). G.f.: x/(1-5*x+4*x^2). - Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Apr 24 2001
%F A002450 a(n)= sum{k=0..n-1, 4^k} a(n)= A001045(2n). - Paul Barry (pbarry(AT)wit.ie), Mar 17 2003
%F A002450 Second binomial transform of A001045. - Paul Barry (pbarry(AT)wit.ie), Mar 28 2003
%F A002450 E.g.f. (exp(4x)-exp(x))/3 - Paul Barry (pbarry(AT)wit.ie), Mar 28 2003
%F A002450 a(0) = 0, a(n+1) = 4*a(n) + 1 . - DELEHAM Philippe (kolotoko(AT)lagoon.nc), Feb 25 2004
%p A002450 [seq((4^n-1)/3,n=0..40)];
%Y A002450 a(n) = (A007583(n)-1)/2.
%Y A002450 Partial sums of powers of 4, A000302.
%Y A002450 a(n)=A000975(2n)/2.
%Y A002450 A084160(n) = 2*a(n).
%Y A002450 Cf. A002446, A024036, A084180, A080674, A047849.
%Y A002450 Sequence in context: A026027 A002054 A028948 this_sequence A084241 A026855 A012814
%Y A002450 Adjacent sequences: A002447 A002448 A002449 this_sequence A002451 A002452 A002453
%K A002450 nonn,easy,nice
%O A002450 0,3
%A A002450 njas
%I A084241
%S A084241 0,1,5,21,85,341,1365,5461,21845,87381,349525,1398101,5592405,22369621,
%T A084241 89478485,357913941,1431655765,5726623061,22906492245,91625968981,
%U A084241 366503875925,1466015503701,5864062014805,23456248059221,93824992236885
%V A084241 0,1,-5,21,-85,341,-1365,5461,-21845,87381,-349525,1398101,-5592405,22369621,-89478485,
%W A084241 357913941,-1431655765,5726623061,-22906492245,91625968981,-366503875925,1466015503701,
%X A084241 -5864062014805,23456248059221,-93824992236885
%N A084241 a(n)=-5a(n-1)-4a(n-2), a(0)=0,a(1)=1.
%C A084241 abs(a(n))=A002450(n)=A001045(2n). Binomial transform is (0,1,-3,9,-27,...).
%F A084241 a(n)=((-1)^n-(-4)^n)/3; a(n)=sum{k=1..n, (-1)^(n+k)binomial(n,k)(-3)^(k-1) }; G.f.: x/((1+x)(1+4x)); E.g.f.: (exp(-x)-exp(-4x))/3.
%Y A084241 Cf. A084240.
%Y A084241 Sequence in context: A002054 A028948 A002450 this_sequence A026855 A012814 A039919
%Y A084241 Adjacent sequences: A084238 A084239 A084240 this_sequence A084242 A084243 A084244
%K A084241 easy,sign
%O A084241 0,3
%A A084241 Paul Barry (pbarry(AT)wit.ie), May 21 2003
Transformation T041 gave a match with:
%I A084240
%S A084240 1,0,4,20,84,340,1364,5460,21844,87380,349524,1398100,5592404,22369620,89478484,
%T A084240 357913940,1431655764,5726623060,22906492244,91625968980,366503875924,1466015503700,
%U A084240 5864062014804,23456248059220,93824992236884,375299968947540
%V A084240 1,0,-4,20,-84,340,-1364,5460,-21844,87380,-349524,1398100,-5592404,22369620,-89478484,
%W A084240 357913940,-1431655764,5726623060,-22906492244,91625968980,-366503875924,1466015503700,
%X A084240 -5864062014804,23456248059220,-93824992236884,375299968947540
%N A084240 a(n)=-5a(n-1)-4a(n-2), a(0)=1, a(1)=0.
%C A084240 A Jacobsthal related sequence.
%C A084240 Inverse binomial transform of A084246. (-1)^n(1-A002450(n)).
%F A084240 a(n)=4*(-1)^n/3-(-4)^n/3; G.f.: (1+5x)/((1+x)(1+4x)); E.g.f.: (4*exp(-x)-exp(-4*x))/3.
%F A084240 a(n)=A084241(n)+(-1)^n a(n)=(-1)^(n+1)(A002450(n)-1)=(-1)^(n+1)(A001045(2n)-1).
%F A084240 a(n)=(4(-1)^n-(-4)^n)/3; G.f.: (1+5x)/((1+x)(1+4x)).
%Y A084240 Cf. A001045.
%Y A084240 Sequence in context: A074358 A055296 A003489 this_sequence A080674 A027156 A017964
%Y A084240 Adjacent sequences: A084237 A084238 A084239 this_sequence A084241 A084242 A084243
%K A084240 easy,sign
%O A084240 0,3
%A A084240 Paul Barry (pbarry(AT)wit.ie), May 23 2003
%I A080674
%S A080674 0,4,20,84,340,1364,5460,21844,87380,349524,1398100,5592404,22369620,89478484,
%T A080674 357913940,1431655764,5726623060,22906492244,91625968980,366503875924,1466015503700,
%U A080674 5864062014804,23456248059220,93824992236884,375299968947540,1501199875790164
%N A080674 (4/3)*(4^n-1).
%Y A080674 a(n) = 2 * A084180(n) = A002450(n+1)-1 = 4 * A002450(n).
%Y A080674 Sequence in context: A055296 A003489 A084240 this_sequence A027156 A017964 A017965
%Y A080674 Adjacent sequences: A080671 A080672 A080673 this_sequence A080675 A080676 A080677
%K A080674 nonn
%O A080674 0,2
%A A080674 njas, Mar 02 2003
Transformation T019 gave a match with:
%I A002001
%S A002001 1,3,12,48,192,768,3072,12288,49152,196608,786432,3145728,
%T A002001 12582912,50331648,201326592,805306368,3221225472,12884901888,
%U A002001 51539607552,206158430208,824633720832,3298534883328,13194139533312
%N A002001 a(n) = 3*4^(n-1), n>0; a(0)=1.
%C A002001 Second binomial transform of (1,1,4,4,16,16,...)=(3*2^n+(-2)^n)/4. - Paul Barry (pbarry(AT)wit.ie), Jul 16 2003
%H A002001 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 456
%F A002001 a(n)=(3*4^n+0^n)/4 (with 0^0=1). E.g.f. (3exp(4x)+1)/4. - Paul Barry (pbarry(AT)wit.ie), Apr 20 2003
%F A002001 With interpolated zeros, this has e.g.f (3*cosh(2x)+1)/4, and binomial transform A006342. - Paul Barry (pbarry(AT)wit.ie), Sep 03 2003
%F A002001 a(n)=sum{j=0..1, sum{k=0..n, C(2n+j,2k) }} - Paul Barry (pbarry(AT)wit.ie), Nov 29 2003
%F A002001 G.f. : (1-x)/(1-4x). The sequence 1,3,-12,48,-192... has g.f. (1+7x)/(1+4x) - Paul Barry (pbarry(AT)wit.ie), Feb 12 2004
%Y A002001 First difference of 4^n (A000302).
%Y A002001 Sequence in context: A088132 A064562 A077828 this_sequence A058371 A060113 A037758
%Y A002001 Adjacent sequences: A001998 A001999 A002000 this_sequence A002002 A002003 A002004
%K A002001 nonn,easy
%O A002001 0,2
%A A002001 njas
Transformation T001 gave a match with:
%I A002450 M3914 N1608
%S A002450 0,1,5,21,85,341,1365,5461,21845,87381,349525,1398101,5592405,22369621,
%T A002450 89478485,357913941,1431655765,5726623061,22906492245,91625968981,
%U A002450 366503875925,1466015503701,5864062014805,23456248059221,93824992236885
%N A002450 (4^n - 1)/3.
%C A002450 For n>0, a(n) is the degree (n-1) "numbral" power of 5 (see A048888 for the definition of numbral arithmetic). Example: a(3)=21, since the numbral square of 5 is 5(*)5 = 101(*)101(base 2) = 101 OR 10100 = 10101(base 2) = 21, where the OR is taken bitwise. - John W. Layman (layman(AT)math.vt.edu), Dec 18 2001
%C A002450 a(n) is composite for all n > 2 and has factors x, (3x+2(-1)^n) where x belongs to A001045. In binary the terms are 1, 101, 10101, 1010101, etc. - John McNamara (mistermac39(AT)yahoo.com), Jan 16 2002
%C A002450 Number of n X 2 binary arrays with path of adjacent 1's from upper left corner to right column. - Ron Hardin (rhh(AT)cadence.com), Mar 16 2002
%C A002450 Collatz-function iteration started at with a[n] will surely ended by 1 in exactly 2n steps. - Labos E. (labos(AT)ana1.sote.hu), Sep 30 2002
%C A002450 Also sum of squares of divisors of 2^n: a(n)=A001157[A000079(n)] - Paul Barry (pbarry(AT)wit.ie), Apr 11 2003
%C A002450 All members of sequence are also generalized octagonal numbers (A001082). - Matthew Vandermast (ghodges14(AT)msn.com), Apr 10 2003
%C A002450 Binomial transform of A000244 (with leading zero) - Paul Barry (pbarry(AT)wit.ie), Apr 11 2003
%D A002450 A. Fletcher, J. C. P. Miller, L. Rosenhead, and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 112.
%D A002450 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
%D A002450 T. N. Thiele, Interpolationsrechnung. Teubner, Leipzig, 1909, p. 35.
%H A002450 H. Bottomley, Illustration of initial terms
%H A002450 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 373
%H A002450 E. W. Weisstein, Link to a section of The World of Mathematics.
%F A002450 a(n+1)= sum(A060921(n,m),m=0..n). G.f.: x/(1-5*x+4*x^2). - Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Apr 24 2001
%F A002450 a(n)= sum{k=0..n-1, 4^k} a(n)= A001045(2n). - Paul Barry (pbarry(AT)wit.ie), Mar 17 2003
%F A002450 Second binomial transform of A001045. - Paul Barry (pbarry(AT)wit.ie), Mar 28 2003
%F A002450 E.g.f. (exp(4x)-exp(x))/3 - Paul Barry (pbarry(AT)wit.ie), Mar 28 2003
%F A002450 a(0) = 0, a(n+1) = 4*a(n) + 1 . - DELEHAM Philippe (kolotoko(AT)lagoon.nc), Feb 25 2004
%p A002450 [seq((4^n-1)/3,n=0..40)];
%Y A002450 a(n) = (A007583(n)-1)/2.
%Y A002450 Partial sums of powers of 4, A000302.
%Y A002450 a(n)=A000975(2n)/2.
%Y A002450 A084160(n) = 2*a(n).
%Y A002450 Cf. A002446, A024036, A084180, A080674, A047849.
%Y A002450 Sequence in context: A026027 A002054 A028948 this_sequence A084241 A026855 A012814
%Y A002450 Adjacent sequences: A002447 A002448 A002449 this_sequence A002451 A002452 A002453
%K A002450 nonn,easy,nice
%O A002450 0,3
%A A002450 njas
%I A084241
%S A084241 0,1,5,21,85,341,1365,5461,21845,87381,349525,1398101,5592405,22369621,
%T A084241 89478485,357913941,1431655765,5726623061,22906492245,91625968981,
%U A084241 366503875925,1466015503701,5864062014805,23456248059221,93824992236885
%V A084241 0,1,-5,21,-85,341,-1365,5461,-21845,87381,-349525,1398101,-5592405,22369621,-89478485,
%W A084241 357913941,-1431655765,5726623061,-22906492245,91625968981,-366503875925,1466015503701,
%X A084241 -5864062014805,23456248059221,-93824992236885
%N A084241 a(n)=-5a(n-1)-4a(n-2), a(0)=0,a(1)=1.
%C A084241 abs(a(n))=A002450(n)=A001045(2n). Binomial transform is (0,1,-3,9,-27,...).
%F A084241 a(n)=((-1)^n-(-4)^n)/3; a(n)=sum{k=1..n, (-1)^(n+k)binomial(n,k)(-3)^(k-1) }; G.f.: x/((1+x)(1+4x)); E.g.f.: (exp(-x)-exp(-4x))/3.
%Y A084241 Cf. A084240.
%Y A084241 Sequence in context: A002054 A028948 A002450 this_sequence A026855 A012814 A039919
%Y A084241 Adjacent sequences: A084238 A084239 A084240 this_sequence A084242 A084243 A084244
%K A084241 easy,sign
%O A084241 0,3
%A A084241 Paul Barry (pbarry(AT)wit.ie), May 21 2003
Transformation T040 gave a match with:
%I A047849
%S A047849 1,2,6,22,86,342,1366,5462,21846,87382,349526,1398102,5592406,22369622,
%T A047849 89478486,357913942,1431655766,5726623062,22906492246,91625968982,
%U A047849 366503875926,1466015503702,5864062014806,23456248059222
%N A047849 a(n)=T(1,n), array T given by A047848.
%F A047849 a(n) =(4^n+2)/3 =4a(n-1)-2 =5a(n-1)-4a(n-2) =2*A007583(n-1) =A002450(n)+1 - Henry Bottomley (se16(AT)btinternet.com), Aug 29 2000
%F A047849 With interpolated zeros, this is (-2)^n/6+2^n/6+(-1)^n/3+1/3. - Paul Barry (pbarry(AT)wit.ie), Aug 26 2003
%F A047849 a(n) = A007583(n) - A002450(n) = A001045(2n+1) - A001045(2n) . - DELEHAM Philippe (kolotoko(AT)lagoon.nc), Feb 25 2004
%Y A047849 n-th difference of a(n), a(n-1), ..., a(0) is 3^(n-1) for n=1, 2, 3, ...
%Y A047849 Sequence in context: A079105 A079104 A029759 this_sequence A073075 A032351 A049135
%Y A047849 Adjacent sequences: A047846 A047847 A047848 this_sequence A047850 A047851 A047852
%K A047849 nonn
%O A047849 0,2
%A A047849 Clark Kimberling (ck6(AT)evansville.edu)
Transformation T030 gave a match with:
%I A003947
%S A003947 1,5,20,80,320,1280,5120,20480,81920,327680,1310720,5242880,
%T A003947 20971520,83886080,335544320,1342177280,5368709120,21474836480,
%U A003947 85899345920,343597383680,1374389534720,5497558138880,21990232555520
%N A003947 Coordination sequence for infinite tree with valency 5.
%H A003947 Index entries for sequences related to trees
%H A003947 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 306
%F A003947 Binomial transform of A060925. Its binomial transform is A003463 (without leading zero). - Paul Barry (pbarry(AT)wit.ie), May 19 2003
%F A003947 a(n)=(5*4^n-0^n)/4; G.f. : (1+x)/(1-4x); E.g.f.: (5exp(4x)-exp(0))/4. - Paul Barry (pbarry(AT)wit.ie), May 19 2003
%p A003947 k:=5; if n = 0 then 1 else k*(k-1)^(n-1); fi;
%Y A003947 Sequence in context: A079737 A028814 A079820 this_sequence A033131 A022021 A030520
%Y A003947 Adjacent sequences: A003944 A003945 A003946 this_sequence A003948 A003949 A003950
%K A003947 nonn
%O A003947 0,2
%A A003947 njas
List of transformations used:
T001 the sequence itself
T004 sequence divided by the gcd of its elements, from the 2nd term
T005 sequence divided by the gcd of its elements, from the 3rd term
T011 sequence 2*u[j]
T012 sequence 3*u[j]
T018 sequence u[j+1]-u[j]
T019 sequence u[j+2]-2*u[j+1]+u[j]
T020 sequence u[j+3]-3*u[j+2]-3*u[j+1]+u[j]
T024 sequence u[j]+u[j+1]
T030 sequence u[j+2]-u[j]
T040 sequence u[j]+1
T041 sequence u[j]-1
Abbreviations used in the above list of transformations:
u[j] = j-th term of the sequence
v[j] = u[j]/(j-1)!
Sn(z) = ordinary generating function
En(z) = exponential generating function
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