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
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).
Programs
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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
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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 *) -
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
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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
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).
Comments