A098523 Expansion of (1+x^2)/(1-x-x^5) = (1+x^2)/((1-x+x^2)*(1-x^2-x^3)).

1, 1, 2, 2, 2, 3, 4, 6, 8, 10, 13, 17, 23, 31, 41, 54, 71, 94, 125, 166, 220, 291, 385, 510, 676, 896, 1187, 1572, 2082, 2758, 3654, 4841, 6413, 8495, 11253, 14907, 19748, 26161, 34656, 45909, 60816, 80564, 106725, 141381, 187290, 248106

Offset: 0

Author

Paul Barry, Sep 12 2004

Keywords

• easy
• nonn

Comments

The expansion of (1+kx^2)/(1-x-k^2*x^5) satisfies the recurrence a(n)=a(n-1)+k^2*a(n-5),a(0)=1,a(1)=1,a(2)=k+1,a(3)=k+1,a(4)=k+1, with a(n)=sum{k=0..floor(n/2), binomial(n-2k,floor(k/2))r^k}.

Links

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

Crossrefs

Cf. A097333.

Programs

Mathematica

CoefficientList[Series[(1+x^2)/(1-x-x^5),{x,0,50}],x] (* or *) LinearRecurrence[ {1,0,0,0,1},{1,1,2,2,2},50] (* Harvey P. Dale, Mar 05 2014 *)

Formula

a(n)=a(n-1)+a(n-5); a(n)=sum{k=0..floor(n/2), binomial(n-2k, floor(k/2))}.

7*a(n) = 8*A182097(n) +5*A182097(n-1) +3*A182097(n-2) - A010892(n) +3*A010892(n-1). - R. J. Mathar, Jul 07 2023