A054894 a(n+1) = 4*a(n) + 4*a(n-1) - 4*a(n-2) - a(n-3) with a(1)=1, a(2)=2, a(3)=11, a(4)=48.

1, 2, 11, 48, 227, 1054, 4921, 22944, 107017, 499106, 2327795, 10856592, 50634107, 236152510, 1101392305, 5136786240, 23957470033, 111735303362, 521122556315, 2430464772336, 11335450631123, 52867436085214, 246568565219689, 1149971737922784, 5363356017597913, 25014169325118818

Offset: 1

Author

Barry Cipra, Jul 04 2000

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Svenja Huntemann, Neil A. McKay, Counting Domineering Positions, arXiv:1909.12419 [math.CO], 2019.
• Index entries for linear recurrences with constant coefficients, signature (4,4,-4,-1).

Crossrefs

Column 3 of A332862 (apart from the initial term 1 here).

Cf. A055518, A055519.

Programs

GAP

a:=[1,2,11,48];; for n in [5..40] do a[n]:=4*(a[n-1]+a[n-2]-a[n-3]) -a[n-4]; od; a; # G. C. Greubel, Dec 29 2019

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( x*(1-2*x-x^2)/(1-4*x-4*x^2+4*x^3+x^4) )); // G. C. Greubel, Dec 29 2019

Maple

seq(coeff(series(x^4/((1+2*x)*(2*x^3+x^2-2*x+1)), x, n+1), x, n), n = 0..40); # G. C. Greubel, Dec 29 2019

Mathematica

CoefficientList[Series[(1-2*x-x^2)/(1-4*x-4*x^2+4*x^3+x^4), {x, 0, 40}], x] (* Vincenzo Librandi, May 05 2013 *)
LinearRecurrence[{4,4,-4,-1},{1,2,11,48},40] (* Harvey P. Dale, May 28 2014 *)

PARI

my(x='x+O('x^66)); Vec(x*(1-2*x-x^2)/(1-4*x-4*x^2+4*x^3+x^4)) \\ Joerg Arndt, May 06 2013

Sage

def A054894_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1-2*x-x^2)/(1-4*x-4*x^2+4*x^3+x^4) ).list()
a=A054894_list(40); a[1:] # G. C. Greubel, Dec 29 2019

Formula

a(n) = Sum_{k=1..n} Fibonacci(k)^3*a(n-k), with a(0)=1. - Vladeta Jovovic, Apr 23 2003

G.f.: x*(1-2*x-x^2)/(1-4*x-4*x^2+4*x^3+x^4). - Vaclav Kotesovec, Nov 27 2012