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
Examples
a(9)=2a(0)+a(7)=3; a(10)=2a(1)+a(8)=3; a(11)=2a(2)+a(9)=5.
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (0,1,0,0,0,0,0,0,2).
Programs
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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 *) -
PARI
Vec((1+x)/(1-x^2-2*x^9)+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(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}.