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<a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5, -7, 5).

G.f.: -((3 (1 - 2 *x + 5 2*x)^2)/(-1 - 5*x + 4 7*x)^2 - 5*x^3).

a(n) = (1/2)*A291337A291005(n) for n >= 0.

(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-2*x+2*x^2)/(1-5*x+7*x^2-5*x^3) )); // G. C. Greubel, Jun 01 2023

(SageMath)

def A291337_list(prec):

P.<x> = PowerSeriesRing(ZZ, prec)

return P( (1-2*x+2*x^2)/(1-5*x+7*x^2-5*x^3) ).list()

A291337_list(30) # G. C. Greubel, Jun 01 2023

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z = 60; s = -((2 (1 - 2 x + 2 x^2))/(-1 + 5 x s - 7 x^2 + 5 xs^3));

u / 2 (* A291337 *)

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allocated for Clark Kimberlingp-INVERT of (1,1,1,1,1,...), where p(S) = 1 - 2 S - 2 S^3.

1, 3, 10, 34, 115, 387, 1300, 4366, 14665, 49263, 165490, 555934, 1867555, 6273687, 21075220, 70798066, 237832225, 798950763, 2683918570, 9016098634, 30287816995, 101745987387, 341795711140, 1148195728966, 3857138603785, 12957301471863, 43527515777650

0,2

Clark Kimberling, <a href="/A291337/b291337.txt">Table of n, a(n) for n = 0..1000</a>

<a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5, -7, 5)

G.f.: -((3 (-2 + 5 x))/(-1 + 4 x)^2).

a(n) = 5*a(n-1) - 7*a(n-2) + 5*a(n-3) for n >= 4.

a(n) = 2*A291337(n) for n >= 0.

z = 60; s = -((2 (1 - 2 x + 2 x^2))/(-1 + 5 x - 7 x^2 + 5 x^3));

Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000012 *)

u = Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A291005 *)

u / 2 (*A291337)

Cf. A000012, A289780, A291000, A291005.

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nonn,easy

Clark Kimberling, Aug 23 2017

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allocated for Clark Kimberling

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