A087432 -id:A087432 - cp's OEIS frontend

A087433 Expansion of g.f.: (1-2x)(1-4x+x^2)/((1-x)(1-3x)(1-4*x)).

1, 2, 6, 20, 70, 252, 926, 3460, 13110, 50252, 194446, 758100, 2973350, 11716252, 46333566, 183739940, 730176790, 2906358252, 11582386286, 46200404980, 184414199430, 736494536252, 2942491360606, 11759505089220, 47006639297270

Offset: 0

Paul Barry, Sep 02 2003

Binomial transform of A087432. a(n+1) = 2*A085282(n).

Counts closed walks of length 2n at a vertex of the cyclic graph on 12 nodes C_12. - Herbert Kociemba, Jun 06 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-19,12).

Cf. A085282, A087432.

[0^n/6+1/3+3^n/3+4^n/6: n in [0..30]]; // Vincenzo Librandi, Aug 12 2011

CoefficientList[Series[(1-2x)(1-4x+x^2)/((1-x)(1-3x)(1-4x)),{x,0,30}],x] (* Harvey P. Dale, Nov 26 2014 *)

G.f.: (1-2*x)*(1-4*x+x^2)/((1-x)*(1-3*x)*(1-4*x)).
a(n) = 0^n/6 + 1/3 + 3^n/3 + 4^n/6.
a(n) = 8*a(n-1) - 19*a(n-2) + 12*a(n-3). - Wesley Ivan Hurt, Jul 11 2023

Definition corrected by Herbert Kociemba, Jun 06 2004

A093379 Expansion of x(1-2x-2x^2)/((1+x)(1-2x)(1-3x)).

Original entry on oeis.org

0, 1, 2, 5, 12, 31, 82, 225, 632, 1811, 5262, 15445, 45652, 135591, 404042, 1206665, 3609072, 10805371, 32372422, 97029885, 290914892, 872395151, 2616486402, 7848061105, 23541387112, 70618568931, 211844521982, 635511196325

Offset: 0

Paul Barry, Apr 28 2004

Binomial transform is A085280, with leading zero.

Index entries for linear recurrences with constant coefficients, signature (4,-1,-6).

Cf. A087432.

CoefficientList[Series[x (1-2x-2x^2)/((1+x)(1-2x)(1-3x)),{x,0,30}],x] (* or *) LinearRecurrence[{4,-1,-6},{0,1,2,5},40] (* Harvey P. Dale, May 10 2019 *)

a(n)=3^n/12-(-1)^n/12+2^n/3-0^n/3;
a(n)=4a(n-1)-a(n-2)-6a(n-3).
a(n)=A015518(n)/3+A000079(n)/3-0^n/3.

cp's OEIS Frontend

A087433 Expansion of g.f.: (1-2x)(1-4x+x^2)/((1-x)(1-3x)(1-4*x)).

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A093379 Expansion of x(1-2x-2x^2)/((1+x)(1-2x)(1-3x)).

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cp's OEIS Frontend

A087433 Expansion of g.f.: (1-2*x)*(1-4*x+x^2)/((1-x)*(1-3*x)*(1-4*x)).

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Magma

Mathematica

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A093379 Expansion of x(1-2x-2x^2)/((1+x)(1-2x)(1-3x)).

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A087433 Expansion of g.f.: (1-2x)(1-4x+x^2)/((1-x)(1-3x)(1-4*x)).