a(n) = 17*a(n-1) + a(n-2) with a(-1) = 0, a(0) = 1.
G.f.: 1/(1 - 17*x - x^2).
E.g.f.: exp(17*x/2)*sinh(sqrt(293)*x/2)/(sqrt(293)/2).
a(n) = ( (17+sqrt(17^2+4))^(n+1) - (17-sqrt(17^2+4))^(n+1) )/(2^(n+1)*sqrt(17^2+4)).
a(n) = (Sum_{i=0..floor(n/2)} binomial(n+1,2*i+1)*17^(n-2*i)*293^i)/2^n.
a(n) = Fibonacci(n+1,17), the (n+1)-th Fibonacci polynomial evaluated at x=17.
a(n) = U(n, 17*i/2)*(-i)^n with i^2=(-1) and U(n, x/2)=S(n, x), see
A049310.
a(n-r-1)*a(n+r-1) - a(n-1)^2 + (-1)^(n-r)*a(r-1)^2 = 0; a(-1) = 0 and n >= r+1.
a(n-1) + a(n+1) =
A090306(n+1);
A090306(n)^2 - 293*a(n-1)^2 - 4*(-1)^n = 0.
a(p-1) == 293^((p-1)/2) (mod p) for odd primes p.
a(2n+1) = 17*
A098248(n) (S(n,291)), a(2n) =
A098250(n) (first differences of S(n,291)).
Limit_{k -> oo} a(n+k)/a(k) = (
A090306(n) + a(n-1)*sqrt(293))/2.
Limit_{n -> oo}
A090306(n)/a(n-1) = sqrt(293).
Sum_{n>=0} (-1)^n/(a(n)*a(n+1)) = (sqrt(293)-17)/2. -
Amiram Eldar, Apr 05 2026
