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.

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A026097 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is an integer, s(0) = 0, |s(i) - s(i-1)| = 1 for i = 1,2,3; |s(i) - s(i-1)| <= 1 for i >= 4. Also a(n) = sum of numbers in row n+1 of the array T defined in A026082 and a(n) = 24*3^(n-4) for n >= 4.

Original entry on oeis.org

1, 2, 4, 8, 24, 72, 216, 648, 1944, 5832, 17496, 52488, 157464, 472392, 1417176, 4251528, 12754584, 38263752, 114791256, 344373768, 1033121304, 3099363912, 9298091736, 27894275208, 83682825624, 251048476872, 753145430616, 2259436291848
Offset: 0

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Author

Clark Kimberling

Keywords

Comments

Also length of successive strings generated by an alternating Kolakoski (2,4) rule starting at 4 (i.e. string begins with 2 if previous string ends with 4 and vice et versa) : 4-->2222-->44224422-->444422224422444422224422-->... and length of strings are 1,4,8,24,72,... - Benoit Cloitre, Oct 15 2005
Also number of words of length n over alphabet {1,2,3} with no fixed points (a fixed point is value i in position i). - Margaret Archibald, Jun 23 2020

References

  • M. Archibald, A. Blecher and A. Knopfmacher, Fixed points in compositions and words, accepted by the Journal of Integer Sequences.

Links

Crossrefs

Essentially the same as A005051.

Programs

Formula

a(n) = 3*a(n-1) for n>3. G.f.: (4*x^3+2*x^2+x-1) / (3*x-1). - Colin Barker, Jun 15 2013
a(n) = floor( (4*n-2)/(n+1) )*a(n-1). Without the floor function the recursion gives the Catalan numbers (A000108). - Hauke Woerpel, Oct 16 2020

A027319 a(n) = Sum_{k=0..m} (k+1) * A026082(n, k), where 0 <= k <= m, m=n for n=0,1,2,3; m=2n for n >= 4.

Original entry on oeis.org

1, 3, 8, 20, 120, 432, 1512, 5184, 17496, 58320, 192456, 629856, 2047032, 6613488, 21257640, 68024448, 216827928, 688747536, 2181033864, 6887475360, 21695547384, 68186006064, 213856109928, 669462604992, 2092070640600, 6527260398672, 20334926626632
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

Or, a(n) = Sum_{k=0..m} (k+1)*T(n,m-k), m=n for n=0,1,2,3; m=2n for n >= 4; T given by A026082.

Links

Programs

  • Mathematica
    CoefficientList[Series[(1 - 3 x - x^2 - x^3 + 72 x^4 - 108 x^5)/(1 - 3 x)^2, {x, 0, 26}], x] (* Michael De Vlieger, Feb 17 2016 *)
  • PARI
    Vec((1-3*x-x^2-x^3+72*x^4-108*x^5)/(1-3*x)^2 + O(x^30)) \\ Colin Barker, Feb 17 2016

Formula

For n>3, a(n) = 8*(n+1)*3^(n-3).
From Colin Barker, Feb 17 2016: (Start)
a(n) = 6*a(n-1) - 9*a(n-2) for n>5.
G.f.: (1 - 3*x - x^2 - x^3 + 72*x^4 - 108*x^5) / (1-3*x)^2.
(End)

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 05 2007

A026083 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is an integer, s(0) = 0 = s(n), |s(i) - s(i-1)| = 1 for i = 1,2,3; |s(i) - s(i-1)| <= 1 for i >= 4. Also a(n) = T(n,n), where T is the array defined in A026082.

Original entry on oeis.org

6, 12, 38, 104, 300, 856, 2464, 7104, 20550, 59580, 173118, 503960, 1469546, 4291644, 12550290, 36746592, 107712306, 316050372, 928224594, 2728494360, 8026707864, 23630376000, 69614498268, 205212650272, 605292727450, 1786351811556
Offset: 4

Views

Author

Clark Kimberling

Keywords

Comments

Third differences of the central trinomial numbers (A002426). - T. D. Noe, Mar 16 2005

Crossrefs

Equals 2 * A024998(n-1). First differences of A024997.

Formula

Conjecture: n*a(n) +(-3*n+5)*a(n-1) +(-n-6)*a(n-2) +3*(n-5)*a(n-3)=0. - R. J. Mathar, Jun 22 2013

A026094 a(n) = T(n,[ n/2 ]), where T is the array defined in A026082.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 16, 28, 93, 156, 518, 864, 2932, 4861, 16709, 27600, 95849, 157899, 552749, 908768, 3201848, 5256009, 18616751, 30523044, 108592239, 177865347, 635175153, 1039511496, 3724193120, 6090715215, 21881993115, 35765791320, 128810517855
Offset: 0

Views

Author

Clark Kimberling

Keywords

A027315 Self-convolution of array T given by A026082.

Original entry on oeis.org

1, 2, 6, 20, 90, 692, 5774, 48924, 417440, 3580036, 30831756, 266463804, 2309848194, 20074999500, 174867895438, 1526259877412, 13344806416950, 116863477094660, 1024844657752730, 8998915251701628, 79108591391227406, 696167664755023388, 6132277201334483178
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

Equivalently, sum of squares of numbers in row n of array T given by A026082.

Extensions

More terms from Sean A. Irvine, Oct 27 2019

A026084 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is an integer, s(0) = 0, |s(i) - s(i-1)| = 1 for i = 1,2,3; |s(i) - s(i-1)| <= 1 for i >= 4, s(n) = 1. Also a(n) = T(n,n-1), where T is the array defined in A026082.

Original entry on oeis.org

3, 13, 33, 98, 278, 804, 2320, 6723, 19515, 56769, 165421, 482793, 1411049, 4129323, 12098151, 35482857, 104169033, 306087111, 900134883, 2649106752, 7801834068, 22992061134, 67799076002, 200040038589, 590529542053, 1744148984223
Offset: 4

Views

Author

Clark Kimberling

Keywords

Crossrefs

First differences of A024998.

Formula

Conjecture: (n+1)*a(n) +3*(-n+1)*a(n-1) +(-n-9)*a(n-2) +3*(n-5)*a(n-3)=0. - R. J. Mathar, Jun 23 2013

A026085 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is an integer, s(0) = 0, |s(i) - s(i-1)| = 1 for i = 1,2,3; |s(i) - s(i-1)| <= 1 for i >= 4, s(n) = 2. Also a(n) = T(n,n-2), where T is the array defined in A026082.

Original entry on oeis.org

4, 8, 27, 76, 226, 660, 1939, 5688, 16704, 49072, 144254, 424296, 1248728, 3677184, 10834416, 31939584, 94205772, 277997400, 820747275, 2424232956, 7163519202, 21176638868, 62626464319, 185276853192, 548326714720, 1623325361424
Offset: 4

Views

Author

Clark Kimberling

Keywords

Crossrefs

First differences of A026069.

Formula

Conjecture: (n+2)*a(n) +(-5*n-1)*a(n-1) +4*(n-4)*a(n-2) +8*(n-1)*a(n-3) +(-5*n+34)*a(n-4) +3*(-n+7)*a(n-5)=0. - R. J. Mathar, Jun 23 2013

A026086 Number of (s(0), s(1), ..., s(n)) such that s(0) = 0, |s(i) - s(i-1)| = 1 for i = 1,2,3; |s(i) - s(i-1)| <= 1 for i >= 4, s(n) = 3; also a(n) = T(n,n-3), where T is the array defined in A026082.

Original entry on oeis.org

1, 6, 16, 52, 156, 475, 1429, 4293, 12853, 38413, 114621, 341639, 1017407, 3027909, 9007017, 26783331, 79622595, 236662764, 703350798, 2090179494, 6211285598, 18457764317, 54851312871, 163009822939, 484469104651, 1439956255806
Offset: 4

Views

Author

Clark Kimberling

Keywords

Crossrefs

First differences of A026070.

Formula

Conjecture: (n+3)*a(n) +5*(-n-1)*a(n-1) +3*(n-5)*a(n-2) +(11*n-9)*a(n-3) +4*(-n+11)*a(n-4) +6*(-n+7)*a(n-5)=0. - R. J. Mathar, Jun 23 2013

A026087 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is an integer, s(0) = 0, |s(i) - s(i-1)| = 1 for i = 1,2,3; |s(i) - s(i-1)| <= 1 for i >= 4, s(n) = 4. Also a(n) = T(n,n-4), where T is the array defined in A026082.

Original entry on oeis.org

1, 2, 9, 28, 93, 294, 925, 2872, 8856, 27136, 82764, 251472, 761774, 2301924, 6941898, 20899680, 62834397, 188690634, 566081421, 1696873148, 5082959517, 15216909686, 45532045749, 136182428520, 407160436435, 1216953379486, 3636353333187
Offset: 4

Views

Author

Clark Kimberling

Keywords

Crossrefs

First differences of A026071.

Formula

Conjecture: -(n-4)*(n+4)*a(n) +(4*n^2-11*n-27)*a(n-1) +(-2*n^2+29*n-24)*a(n-2) -(4*n+5)*(n-5)*a(n-3) +3*(n-5)*(n-6)*a(n-4)=0. - R. J. Mathar, Jun 23 2013

A026088 a(n) = T(2n-1,n), where T is the array defined in A026082.

Original entry on oeis.org

1, 3, 8, 52, 294, 1691, 9736, 56277, 326430, 1899530, 11085360, 64857925, 380331474, 2234804775, 13155328400, 77565394650, 458003596050, 2707970743326, 16030265384752, 94997726453172
Offset: 1

Views

Author

Clark Kimberling

Keywords

Showing 1-10 of 20 results. Next

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

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