A237716 7-distance Pell sequence.

1, 1, 1, 1, 1, 1, 1, 3, 3, 5, 5, 7, 7, 9, 13, 15, 23, 25, 37, 39, 55, 65, 85, 111, 135, 185, 213, 295, 343, 465, 565, 735, 935, 1161, 1525, 1847, 2455, 2977, 3925, 4847, 6247, 7897, 9941, 12807, 15895, 20657, 25589, 33151, 41383, 53033, 66997

Offset: 0

Author

Sergio Falcon, Feb 12 2014

Keywords

• nonn
• easy

Examples

			a(7)=2a(0)+a(5)=3; a(8)=2a(1)+a(6)=3; a(9)=2a(2)+a(7)=5.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,1,0,0,0,0,2).

Crossrefs

Cf. A000129, A122522, A159284, A237714.

Programs

Mathematica

For[j = 0, j < 7, j++, a[j] = 1]
For[j = 7, j < 51, j++, a[j] = 2 a[j - 7] + a[j - 2]]
Table[a[j], {j, 0, 50}]
CoefficientList[Series[(1 + x)/(1 - x^2 - 2 x^7), {x,0,50}], x] (* G. C. Greubel, May 01 2017 *)

PARI

Vec((1+x)/(1-x^2-2*x^7)+O(x^99)) \\ Charles R Greathouse IV, Mar 06 2014

Formula

a(0)=1, a(1)=1, a(2)=1, a(3)=1, a(4)=1, a(5)=1, a(6)=1; a(n) = 2*a(n-7) + a(n-2) for n>=7.

G.f.: (1 + x)/(1 - x^2 - 2*x^7).

a(2*n) = Sum_{j=0..n/7} binomial(n-5*j, 2*j)*2^(2*j) + Sum_{j=0..(n-4)/7} binomial(n-3-5*j, 2*j+1)*2^(2*j+1).

a(2*n+1) = Sum_{j=0..n/7} binomial(n-5*j, 2*j)*2^(2*j) + Sum_{j=0..(n-3)/7} binomial(n-2-5*j, 2*j+1)*2^(2*j+1).