A237718 9-distance Pell numbers.

1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 15, 17, 25, 27, 39, 41, 57, 59, 79, 89, 113, 139, 167, 217, 249, 331, 367, 489, 545, 715, 823, 1049, 1257, 1547, 1919, 2281, 2897, 3371, 4327, 5017, 6425, 7531, 9519

Offset

0,10

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (0,1,0,0,0,0,0,0,2).

Formula

a(0)=1, a(1)=1, a(2)=1, a(3)=1, a(4)=1, a(5)=1, a(6)=1, a(7)=1, a(8)=1; a(n) = 2*a(n-9) + a(n-2) for n>=9.

G.f. (1+x)/(1-x^2-2x^9).

a(2*n) = Sum_{j=0..n/9} Binomial[n-7j, 2j]*2^{2j} + Sum_{j=0..(n-5)/9} Binomial[n-4-7j, 2j+1]*2^{2j+1}.

a(2*n+1) = Sum_{j=0..n/9} Binomial[n-7j, 2j]*2^{2j} + Sum_{j=0..(n-4)/9} Binomial[n-3-7j, 2j+1]*2^{2j+1}.

Example

a(9)=2a(0)+a(7)=3; a(10)=2a(1)+a(8)=3; a(11)=2a(2)+a(9)=5.

Mathematica

For[j = 0, j < 9, j++, a[j] = 1]
For[j = 9, j < 51, j++, a[j] = 2 a[j - 9] + a[j - 2]]
Table[a[j], {j, 0, 50}]
CoefficientList[Series[(1 + x)/(1 - x^2 - 2 x^9), {x, 0, 50}], x] (* G. C. Greubel, May 01 2017 *)

Prog

(PARI) Vec((1+x)/(1-x^2-2*x^9)+O(x^99)) \\ Charles R Greathouse IV, Mar 06 2014

Crossrefs

Cf. A000129, A122522, A159284, A237714, A237716, A237717.

Sequence in context: A339110 A245150 A339109 * A245149 A339108 A206914

Adjacent sequences: A237715 A237716 A237717 * A237719 A237720 A237721

Keyword

nonn,easy

Author

Sergio Falcon, Feb 12 2014

Status

approved