A238391 Expansion of (1+x)/(1-x^2-3*x^5).

1, 1, 1, 1, 1, 4, 4, 7, 7, 10, 19, 22, 40, 43, 70, 100, 136, 220, 265, 430, 565, 838, 1225, 1633, 2515, 3328, 5029, 7003, 9928, 14548, 19912, 29635, 40921, 59419, 84565, 119155, 173470, 241918, 351727, 495613, 709192, 1016023, 1434946, 2071204, 2921785, 4198780, 5969854, 8503618, 12183466, 17268973, 24779806

Offset: 0

Author

Sergio Falcon, Feb 26 2014

Keywords

• nonn
• easy

Examples

			a(5) = 3*a(0)+a(3)=4; a(6) = 3*a(1)+a(4)=4; a(7) = 3*a(2)+a(5)=7.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000 [Terms 0 through 500 were computed by G. C. Greubel]
• Index entries for linear recurrences with constant coefficients, signature (0,1,0,0,3).

Programs

Mathematica

For[j = 0, j < 5, j++, a[j] = 1]; For[j = 5, j < 51, j++, a[j] = 3 a[j - 5] + a[j - 2]]; Table[a[j], {j, 0, 50}]
CoefficientList[Series[(1 + x)/(1 - x^2 - 3 x^5), {x, 0, 50}], x] (* Michael De Vlieger, Jan 27 2016 *)

PARI

Vec((1+x)/(1-x^2-3*x^5) + O(x^50)) \\ Michel Marcus, Jan 27 2016

Formula

a(0)=1, a(1)=1, a(2)=1, a(3)=1, a(4)=1; a(n) = 3*a(n-5)+a(n-2) for n>4.

a(2n) = Sum_{j=0..n/5} binomial(n-3j,2j)*3^(2j) + Sum_{j=0..(n-3)/5} binomial(n-2-3j,2j+1)*3^(2j+1).

a(2n+1) = Sum_{j=0..n/5} binomial(n-3j,2j)*3^{2j} + Sum_{j=0..(n-2)/5} binomial(n-1-3j,2j+1)*3^(2j+1).