A113021 Expansion of x^2/(1 - 2*x + 2*x^2 - x^3 - x^4).

0, 0, 1, 2, 2, 1, 1, 4, 9, 12, 11, 11, 21, 43, 66, 78, 88, 129, 226, 360, 485, 605, 826, 1287, 2012, 2881, 3851, 5239, 7669, 11592, 16936, 23596, 32581, 46498, 68366, 99913, 142173, 199384, 282701, 408720, 593595, 851835, 1207901, 1714447, 2458522

Offset

0,4

Comments

Diagonal sums of number triangle A113020.

Links

Matthew House, Table of n, a(n) for n = 0..6387

Index entries for linear recurrences with constant coefficients, signature (2,-2,1,1).

Formula

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

a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} C(n-k, j)*C(0, j-k)*Fibonacci(j-2k);

a(n) = Sum_{k=0..floor(n/2)} (-1)^(k+1)*binomial(n-k, k)*Fibonacci(k).

Mathematica

Table[Sum[(-1)^(k + 1) Binomial[n - k, k] Fibonacci@ k, {k, 0, Floor[n/2]}], {n, 0, 44}] (* Michael De Vlieger, Feb 13 2017 *)
LinearRecurrence[{2, -2, 1, 1}, {0, 0, 1, 2}, 50] (* Harvey P. Dale, Jul 16 2018 *)

Prog

(PARI) a(n)=([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; 1, 1, -2, 2]^n*[0; 0; 1; 2])[1, 1] \\ Charles R Greathouse IV, Feb 14 2017
(PARI) x='x+O('x^30); concat([0, 0], Vec(x^2/(1-2*x+2*x^2-x^3-x^4))) \\ G. C. Greubel, Apr 09 2018
(Magma) I:=[0, 0, 1, 2]; [n le 4 select I[n] else 2*Self(n-1) - 2*Self(n-2) + Self(n-3) + Self(n-4): n in [1..30]]; // G. C. Greubel, Apr 09 2018

Crossrefs

Cf. A096750, A113020.

Sequence in context: A225201 A124976 A176663 * A298261 A368135 A341991

Adjacent sequences: A113018 A113019 A113020 * A113022 A113023 A113024

Keyword

easy,nonn

Author

Paul Barry, Oct 11 2005

Status

approved