A087433 -id:A087433 - cp's OEIS frontend

A087432 Expansion of 1+x(1-x-4x^2)/((1+x)(1-2x)(1-3x)).

1, 1, 3, 7, 19, 51, 143, 407, 1179, 3451, 10183, 30207, 89939, 268451, 802623, 2402407, 7196299, 21567051, 64657463, 193885007, 581480259, 1744091251, 5231574703, 15693326007, 47077181819, 141225953051, 423666674343

Offset: 0

Paul Barry, Sep 02 2003

Binomial transform of A047849 (with interpolated zeros, 1,0,2,0,6,0,...). Binomial transform is A087433.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-1,-6).

First differences of A093379.

CoefficientList[Series[1+x (1-x-4x^2)/((1+x)(1-2x)(1-3x)),{x,0,30}],x] (* or *) LinearRecurrence[{4,-1,-6},{1,1,3,7},30] (* Harvey P. Dale, Aug 23 2017 *)

Vec((x-1)*(2*x^2+2*x-1)/((1+x)*(1-2*x)*(1-3*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012, corrected Nov 27 2014

a(n) = (-1)^n/6+2^n/3+3^n/6, n>0.

For n>4, a(n) = 6*a(n-1) - 9*a(n-2) - 4*a(n-3) + 12*a(n-4). - Gary W. Adamson, Jun 14 2006

A094659 Number of closed walks of length n at a vertex of the cyclic graph on 7 nodes C_7.

Original entry on oeis.org

1, 0, 2, 0, 6, 0, 20, 2, 70, 18, 252, 110, 924, 572, 3434, 2730, 12902, 12376, 48926, 54264, 187036, 232562, 720062, 980674, 2789164, 4086550, 10861060, 16878420, 42484682, 69242082, 166823430, 282580872, 657178982, 1148548016, 2595874468

Offset: 0

Herbert Kociemba, Jun 06 2004

In general, a(n,m) = (2^n/m)*Sum_{k=0..m-1} cos(2*Pi*k/m)^n gives the number of closed walks of length n at a vertex of the cyclic graph on m nodes C_m.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,4,-3,-2).

Cf. A199572 (m=2), A078008 (m=3), A199573 (m=4), A054877 (m=5), A047849 (bisection of m=6), A063376 (bisection of m=8), A094233 (m=9), A095929 (bisection of m=10), A087433 (bisection of m=12).

f[n_] := FullSimplify[ TrigToExp[ 2^n/7 Sum[Cos[2Pi*k/7]^n, {k, 0, 6}]]]; Table[ f[n], {n, 0, 36}] (* Robert G. Wilson v, Jun 09 2004 *)
LinearRecurrence[{1,4,-3,-2},{1,0,2,0},40] (* Harvey P. Dale, Jun 12 2014 *)

a(n) = (2^n/7)*Sum_{k=0..6} cos(2*Pi*k/7)^n.
a(n) = 7(a(n-2) - 2a(n-4) + a(n-6)) + 2a(n-7).
G.f.: (1-x-2x^2+x^3)/((2x-1)(-1-x+2x^2+x^3)).
a(0)=1, a(1)=0, a(2)=2, a(3)=0, a(n)=a(n-1)+4*a(n-2)-3*a(n-3)-2*a(n-4). - Harvey P. Dale, Jun 12 2014
7*a(n) = 2^n + 2*A094648(n). - R. J. Mathar, Nov 03 2020

More terms from Robert G. Wilson v, Jun 09 2004

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A087432 Expansion of 1+x(1-x-4x^2)/((1+x)(1-2x)(1-3x)).

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A094659 Number of closed walks of length n at a vertex of the cyclic graph on 7 nodes C_7.

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cp's OEIS Frontend

A087432 Expansion of 1+x*(1-x-4*x^2)/((1+x)*(1-2*x)*(1-3*x)).

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A094659 Number of closed walks of length n at a vertex of the cyclic graph on 7 nodes C_7.

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A087432 Expansion of 1+x(1-x-4x^2)/((1+x)(1-2x)(1-3x)).