%I #14 Jul 23 2023 17:08:11

%S 1,1,1,3,3,3,3,7,11,15,27,39,51,63,91,135,195,303,459,663,915,1279,

%T 1819,2599,3811,5647,8299,11959,17075,24351,34747,49991,72579,105775,

%U 153611,221911,319315,458303,658267,948583,1371683,1986127,2873771,4151031

%N Expansion of (1 + 2*x^3)/(1 - x - 4*x^7).

%C The expansion of (1 + k*x^2)/(1 - x - k^2*x^7) satisfies the recurrence a(n) = a(n-1) + k^2*a(n-7), a(0)=1, a(1)=1, a(2)=1, a(3)=k+1, a(4)=k+1, a(5)=k+1, a(6)=k+1 with a(n) = Sum_{k=0..floor(n/3)} binomial(n-3k, floor(k/2))*r^k.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,4).

%F a(n) = a(n-1) + 4*a(n-7);

%F a(n) = Sum_{k=0..floor(n/3)} binomial(n-3k, floor(k/2))*2^k.

%t LinearRecurrence[{1,0,0,0,0,0,4},{1,1,1,3,3,3,3},50] (* _Harvey P. Dale_, Jan 26 2014 *)

%o (PARI) Vec((1+2*x^3)/(1-x-4*x^7)+O(x^66)) \\ _Joerg Arndt_, Jan 28 2014

%Y Cf. A098524.

%K easy,nonn

%O 0,4

%A _Paul Barry_, Sep 12 2004

%E Name corrected by _Harvey P. Dale_, Jan 26 2014