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 1-3 of 3 results.

A077941 Expansion of 1/(1-2*x+x^2+x^3).

Original entry on oeis.org

1, 2, 3, 3, 1, -4, -12, -21, -26, -19, 9, 63, 136, 200, 201, 66, -269, -805, -1407, -1740, -1268, 611, 4230, 9117, 13393, 13439, 4368, -18096, -53999, -94270, -116445, -84621, 41473, 284012, 611172, 896859, 898534, 289037, -1217319, -3622209, -6316136, -7792744, -5647143, 2814594
Offset: 0

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Author

N. J. A. Sloane, Nov 17 2002

Keywords

Comments

With three leading zeros, is the inverse binomial transform of A077868, with three leading zeros. - Paul Barry, Oct 22 2004

Links

Crossrefs

Cf. A077990.

Programs

Formula

a(n) = sum{k=0..n+3, C(n+3, k)(-1)^(n+3-k)*sum{j=0..floor((k-2)/2), C(k-2-2j, j+1)}}. - Paul Barry, Oct 22 2004
a(n) = sum{k=0..floor(n/3), C(n+1-k,n-3k)*(-1)^k}. - Tani Akinari, Oct 10 2014

A107068 Expansion of 1/((1+x)^3-x^4).

Original entry on oeis.org

1, -3, 6, -10, 16, -27, 49, -92, 172, -316, 573, -1035, 1874, -3406, 6204, -11303, 20577, -37432, 68072, -123800, 225193, -409683, 745342, -1355970, 2466760, -4487395, 8163217, -14850196, 27015092, -49145300, 89404037, -162641499, 295872778, -538243174, 979156724, -1781254927, 3240410561
Offset: 0

Views

Author

Paul Barry, May 10 2005

Keywords

Links

Crossrefs

Cf. A077990.

Programs

  • Mathematica
    LinearRecurrence[{-3,-3,-1,1},{1,-3,6,-10},40] (* Harvey P. Dale, Jul 16 2018 *)

Formula

G.f.: 1/(1+3x+3x^2+x^3-x^4); a(n)=sum{k=0..n+4, (-1)^(n-k)*C(n+4, k)*sum{j=0..floor(k/4), C(k-3j, j)}}.

A301772 Number of odd chordless cycles in the n-antiprism graph.

Original entry on oeis.org

0, 2, 0, 2, 8, 2, 24, 16, 48, 92, 100, 310, 344, 808, 1344, 2102, 4480, 6462, 13092, 21662, 37488, 69904, 113652, 212844, 359856, 636402, 1134068, 1937072, 3493120, 6012746, 10639264, 18706394, 32550976, 57727738, 100407848, 177116816, 310493720, 543717148
Offset: 0

Views

Author

Eric W. Weisstein, Mar 26 2018

Keywords

Comments

Sequence extended to a(0)-a(3) using the formula/recurrence (actual 3-antiprism count is 0).

Links

Programs

  • Mathematica
    Table[(RootSum[-1 + #1 - 2 #1^2 + #1^3 &, #1^n &] - RootSum[-1 + #1 + 2 #1^2 + #1^3 &, #1^n &])/2, {n, 0, 20}]
LinearRecurrence[{0, 2, 2, -1, 2, -1}, {0, 2, 0, 2, 8, 2}, 20]
CoefficientList[Series[2 x (1 - x^2 + 2 x^3)/(1 - 2 x^2 - 2 x^3 + x^4 - 2 x^5 + x^6), {x, 0, 20}], x]

Formula

a(n) = 2*a(n-2) + 2*a(n-3) - a(n-4) + 2*a(n-5) - a(n-6).
G.f.: 2*x*(1 - x^2 + 2*x^3)/( (x^3-x^2-2*x-1)*(x^3-x^2+2*x-1)).
2*a(n) = -3*A077990(n) -4*A077990(n-1)-A077990(n-2) +3*A005314(n+1) -4*A005314(n)+A005314(n-1). - R. J. Mathar, Feb 25 2024
Showing 1-3 of 3 results.

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

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