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A120748 Expansion of x^2*(1 + 2*x - x^2)/(1 - x - 3*x^2 - x^3 + x^4). 1

%I #26 Nov 14 2022 02:19:29

%S 0,1,3,5,15,32,79,185,439,1041,2464,5841,13835,32781,77663,184000,

%T 435935,1032817,2446959,5797345,13735104,32541281,77096979,182658581,

%U 432755695,1025287136,2429115823,5755074345,13634953255,32304004977,76534823264,181326717105

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

%C Lim_{n->oo} a(n)/a(n-1) is 2.3692054...; largest real eigenvalue of M and a root of the characteristic polynomial x^4 - x^3 - 3x^2 - x + 1.

%C a(n) is the top left entry of the n-th power of the 4 X 4 matrix M = [0,1,1,0; 1,1,1,0; 0,1,0,1; 1,0,1,0].

%H G. C. Greubel, <a href="/A120748/b120748.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,1,-1).

%F a(n) = a(n-1) + 3*a(n-2) + a(n-3) - a(n-4).

%e a(8) = 439 = a(7) + 3*a(6) + a(5) - a(4) = 185 + 3*79 + 32 - 15.

%t LinearRecurrence[{1,3,1,-1}, {0,1,3,5}, 40] (* _Amiram Eldar_, Feb 28 2020 *)

%o (Magma) I:=[0,1,3,5]; [n le 4 select I[n] else Self(n-1) +3*Self(n-2) +Self(n-3) -Self(n-4): n in [1..41]]; // _G. C. Greubel_, Nov 13 2022

%o (SageMath)

%o def A120748_list(prec):

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

%o return P( x^2*(1+2*x-x^2)/(1-x-3*x^2-x^3+x^4) ).list()

%o a=A120748_list(40); a[1:] # _G. C. Greubel_, Nov 13 2022

%K nonn,easy

%O 1,3

%A _Gary W. Adamson_, Jul 01 2006

%E More terms from _Amiram Eldar_, Feb 28 2020

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Last modified March 28 16:28 EDT 2024. Contains 371254 sequences. (Running on oeis4.)