cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 13 results. Next

A026269 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is a nonnegative integer, s(0) = 0 = s(n), s(1) = 1, |s(i) - s(i-1)| <= 1 for i >= 2, |s(2) - s(1)| = 1, |s(3) - s(2)| = 1 if s(2) = 1. Also a(n) = T(n,n) and a(n) = Sum{T(k,k-1)}, k = 1,2,...,n, where T is array in A026268.

Original entry on oeis.org

1, 2, 4, 10, 25, 64, 166, 436, 1157, 3098, 8360, 22714, 62086, 170614, 471096, 1306374, 3636708, 10159590, 28473132, 80032638, 225562929, 637301652, 1804751718, 5121677512, 14563448593, 41487279622, 118389089432, 338381552294, 968627180975
Offset: 2

Views

Author

Clark Kimberling

Keywords

Comments

Convolution of [1,2,3,6,13,..] (A005554) with [1,0,1,2,5,12...] (essentially A002026). - R. J. Mathar, Nov 01 2021

Links

Crossrefs

First differences of A102071.

Programs

  • Mathematica
    Drop[CoefficientList[Series[4x^2(1-x^2)/(1-x+Sqrt[1-2x-3x^2])^2, {x,0,30}],x],2] (* Harvey P. Dale, May 05 2011 *)

Formula

G.f.: 4z^2(1-z^2)/[1-z+sqrt(1-2z-3z^2)]^2.
D-finite with recurrence (n+2)*a(n) +(-3*n-1)*a(n-1) +(-n+2)*a(n-2) +3*(n-5)*a(n-3)=0. - R. J. Mathar, Jun 10 2013
a(n) ~ 8 * 3^(n-3/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 12 2014
a(n) = A002026(n-1) - A002026(n-3). - R. J. Mathar, Nov 01 2021

Extensions

More terms from Ralf Stephan, Dec 30 2004

A026299 Number of (s(0), s(1), ..., s(n)) such that every s(i) is a nonnegative integer, s(0) = 0, s(1) = 1, |s(i) - s(i-1)| <= 1 for i >= 2, |s(2) - s(1)| = 1, |s(3) - s(2)| = 1 if s(2) = 1. Also sum of numbers in row n+1 of the array T in A026268.

Original entry on oeis.org

1, 3, 7, 19, 53, 149, 422, 1202, 3440, 9884, 28495, 82387, 238801, 693689, 2018981, 5886329, 17187891, 50257299, 147135189, 431245977, 1265264799, 3715761759, 10921722348, 32127865392, 94578844458, 278614855862, 821281118993, 2422356077357, 7148679142639
Offset: 0

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A026268.

Programs

  • Maple
    A026268 := proc(nmax) local T,i,j; T := array(0..nmax,0..nmax) ; for i from 0 to nmax do T[i,0] := 1; od ; T[1,1] := 1 ; if nmax >= 2 then T[2,1] := 1 ; T[2,2] := 1 ; fi ; if nmax >= 3 then T[3,1] := 2 ; T[3,2] := 2 ; T[3,3] := 2 ; fi ; for i from 4 to nmax do T[i,1] := i-1 ; T[i,i] := T[i-1,i-2]+T[i-1,i-1] ; for j from 2 to i-1 do T[i,j] := T[i-1,j-2]+T[i-1,j-1]+T[i-1,j] ; od ; od ; RETURN(T) ; end: A026299 := proc(n) local T ; if n =0 then RETURN(1) ; else T := A026268(n+1) ; sum(T[n+1,i],i=0..n+1) ; fi ; end ; for n from 0 to 30 do printf("%d,",A026299(n)) ; od ; # R. J. Mathar, Oct 31 2006

Formula

Conjecture: (n+2)*a(n) +(-3*n-4)*a(n-1) +(-n+2)*a(n-2) +3*(n-4)*a(n-3)=0. - R. J. Mathar, Jun 23 2013

Extensions

Corrected and extended by R. J. Mathar, Oct 31 2006

A026270 Number of (s(0), s(1), ..., s(n)) such that every s(i) is a nonnegative integer, s(0) = 0, s(1) = 1 = s(n), |s(i) - s(i-1)| <= 1 for i >= 2, |s(2) - s(1)| = 1, |s(3) - s(2)| = 1 if s(2) = 1. Also T(n,n-1), where T is the array in A026268.

Original entry on oeis.org

1, 2, 6, 15, 39, 102, 270, 721, 1941, 5262, 14354, 39372, 108528, 300482, 835278, 2330334, 6522882, 18313542, 51559506, 145530291, 411738723, 1167450066, 3316925794, 9441771081, 26923831029, 76901809810, 219992462862, 630245628681, 1808029517585
Offset: 2

Views

Author

Clark Kimberling

Keywords

Crossrefs

First differences of A026269. Pairwise sums of A026122.

Formula

G.f.: -1 + 4z^2(1-z)(1-z^2)/[1-z+sqrt(1-2z-3z^2)]^2.
Conjecture: (n+3)*a(n) +3*(-n-1)*a(n-1) +(-n-1)*a(n-2) +3*(n-5)*a(n-3)=0. - R. J. Mathar, Jun 23 2013

A026288 Number of (s(0), s(1), ..., s(n)) such that every s(i) is an integer, s(0) = 0, s(1) = 1, s(n) = 2, |s(i) - s(i-1)| <= 1 for i >= 2, |s(2) - s(1)| = 1, |s(3) - s(2)| = 1 if s(2) = 1. Also T(n,n-2), where T is the array in A026268.

Original entry on oeis.org

1, 2, 5, 14, 38, 104, 285, 784, 2164, 5994, 16658, 46442, 129868, 364182, 1023960, 2886174, 8153952, 23086374, 65497653, 186175794, 530148414, 1512174076, 4320093569, 12360382436, 35414530188, 101603373430, 291864076387, 839402336610
Offset: 2

Views

Author

Clark Kimberling

Keywords

Crossrefs

Pairwise sums of A026123.

Formula

G.f.: 8z^2(1-z)(1-z^2)/[1-z+sqrt(1-2z-3z^2)]^3.
D-finite with recurrence: (n+4)*a(n) +(-5*n-11)*a(n-1) +(5*n+2)*a(n-2) +(5*n-13)*a(n-3) +6*(-n+5)*a(n-4)=0. - R. J. Mathar, Jun 23 2013

A026289 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is an integer, s(0) = 0, s(1) = 1, s(n) = 3, |s(i) - s(i-1)| <= 1 for i >= 2, |s(2) - s(1)| = 1, |s(3) - s(2)| = 1 if s(2) = 1. Also a(n) = T(n,n-3), where T is the array in A026268.

Original entry on oeis.org

1, 3, 9, 27, 79, 229, 659, 1889, 5402, 15430, 44054, 125786, 359296, 1026936, 2937444, 8409540, 24097737, 69118635, 198442329, 570286939, 1640469427, 4723363073, 13612376671, 39265012213, 113358893147, 327545797361, 947203621523, 2741308151929, 7939698087777
Offset: 3

Views

Author

Clark Kimberling

Keywords

Crossrefs

Pairwise sums of A026124.

Formula

G.f.: 16*z^3*(1-z)*(1-z^2)/[1-z+sqrt(1-2*z-3*z^2)]^4.

Extensions

a(27) corrected and more terms from Sean A. Irvine, Sep 24 2019

A026290 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is an integer, s(0) = 0, s(1) = 1, s(n) = 4, |s(i) - s(i-1)| <= 1 for i >=2, |s(2) - s(1)| = 1, |s(3) - s(2)| = 1 if s(2) = 1. Also a(n) = T(n,n-4), where T is the array in A026268.

Original entry on oeis.org

1, 4, 14, 46, 145, 446, 1349, 4034, 11966, 35290, 103642, 303458, 886548, 2585922, 7534245, 21934524, 63826041, 185668816, 540034074, 1570719570, 4568920029, 13292253106, 38679350746, 112583530784, 327793747775, 954702193796
Offset: 4

Views

Author

Clark Kimberling

Keywords

Crossrefs

Pairwise sums of A026145.
Pairwise sums of A026125.

Formula

G.f.: 32z^4(1-z)(1-z^2)/[1-z+sqrt(1-2z-3z^2)]^5.

A026291 a(n) = T(2n-1,n), where T is the array in A026268.

Original entry on oeis.org

2, 14, 79, 446, 2530, 14442, 82913, 478384, 2771900, 16119808, 94038087, 550083340, 3225409094, 18951732050, 111562247565, 657813211278, 3884461464740, 22968812720476, 135978792939818
Offset: 2

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A026268.

A026292 a(n) = T(2n,n), where T is the array in A026268.

Original entry on oeis.org

1, 1, 5, 27, 145, 796, 4438, 25034, 142479, 816620, 4706977, 27256963, 158450461, 924129535, 5404960900, 31689099660, 186188338689, 1096001511849, 6462416487070, 38161656273410, 225653893420214, 1335936968552131, 7917871096128978, 46975164555309282, 278951672249192025
Offset: 0

Views

Author

Clark Kimberling

Keywords

Extensions

Corrected and extended by Sean A. Irvine, Sep 24 2019

A026293 a(n) = T(3n,n), where T is the array in A026268.

Original entry on oeis.org

1, 2, 14, 106, 834, 6732, 55289, 459742, 3858524, 32618022, 277326906, 2369003910, 20315879480, 174798617778, 1508225204024, 13045333585950, 113076177711662, 981982906771978, 8542048596893910, 74416362792323370, 649168980344148876, 5669882132589477798
Offset: 0

Views

Author

Clark Kimberling

Keywords

Extensions

a(5) corrected and more terms from Sean A. Irvine, Sep 24 2019

A026294 T(4n,n), where T is the array in A026268.

Original entry on oeis.org

1, 3, 27, 266, 2743, 29068, 313633, 3427358, 37810605, 420193609, 4696967294, 52753182426, 594829939672, 6729488070844, 76349568744149, 868359048628146, 9897485221142239, 113024337104850132, 1292850211089736299, 14810711124377556034, 169898355365054443380
Offset: 0

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A026268.

Extensions

More terms from Sean A. Irvine, Sep 24 2019
Showing 1-10 of 13 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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