A250102 a(n) = 2*5^n - (1+2i)^(2n) - (1-2i)^(2n) where i = sqrt(-1).

0, 16, 64, 16, 2304, 5776, 7744, 309136, 451584, 2062096, 38837824, 27920656, 424030464, 4570300816, 1039933504, 74815378576, 501671890944, 2396689936, 11857885086784, 50863084730896, 8725926945024, 1727825132557456, 4673690093529664, 5056176437385616, 234290415599944704

Offset: 0

Author

N. J. A. Sloane, Nov 15 2014

Keywords

• nonn

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..1430
• Ira M. Gessel, Rational Functions With Nonnegative Integer Coefficients, slides, 50th Séminaire Lotharingien de Combinatoire, 2003.
• Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.
• Index entries for linear recurrences with constant coefficients, signature (-1,5,125).

Crossrefs

Equals 16*A094423.

Programs

Mathematica

Array[2*5^# - (1 + 2 I)^(2 #) - (1 - 2 I)^(2 #) &, 25, 0] (* Michael De Vlieger, Jun 19 2018 *)
LinearRecurrence[{-1,5,125},{0,16,64},30] (* Harvey P. Dale, May 26 2020 *)

PARI

a(n) = 2*5^n - (1+2*I)^(2*n) - (1-2*I)^(2*n) \\ Michel Marcus, Aug 28 2015

PARI

concat(0, Vec(16*x*(1 + 5*x) / ((1 - 5*x)*(1 + 6*x + 25*x^2)) + O(x^30))) \\ Colin Barker, Feb 20 2019

Formula

From Colin Barker, Feb 20 2019: (Start)

G.f.: 16*x*(1 + 5*x) / ((1 - 5*x)*(1 + 6*x + 25*x^2)).

a(n) = -a(n-1) + 5*a(n-2) + 125*a(n-3) for n>2.

(End)