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

0, 1, 3, 5, 15, 32, 79, 185, 439, 1041, 2464, 5841, 13835, 32781, 77663, 184000, 435935, 1032817, 2446959, 5797345, 13735104, 32541281, 77096979, 182658581, 432755695, 1025287136, 2429115823, 5755074345, 13634953255, 32304004977, 76534823264, 181326717105

Offset

1,3

Comments

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.

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].

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (1,3,1,-1).

Formula

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

Example

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

Mathematica

LinearRecurrence[{1, 3, 1, -1}, {0, 1, 3, 5}, 40] (* Amiram Eldar, Feb 28 2020 *)
CoefficientList[Series[x^2(1+2x-x^2)/(1-x-3x^2-x^3+x^4), {x, 0, 40}], x] (* Harvey P. Dale, Mar 27 2026 *)

Prog

(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
(SageMath)
def A120748_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( x^2*(1+2*x-x^2)/(1-x-3*x^2-x^3+x^4) ).list()
a=A120748_list(40); a[1:] # G. C. Greubel, Nov 13 2022

Crossrefs

Sequence in context: A127978 A018470 A281438 * A182143 A193649 A186341

Adjacent sequences: A120745 A120746 A120747 * A120749 A120750 A120751

Keyword

nonn,easy

Author

Gary W. Adamson, Jul 01 2006

Extensions

More terms from Amiram Eldar, Feb 28 2020

Status

approved