A179992 a(n) = a(n-1) + a(n-2) + n^2 for n >= 3, a(1)=2, and a(2)=5.
2, 5, 16, 37, 78, 151, 278, 493, 852, 1445, 2418, 4007, 6594, 10797, 17616, 28669, 46574, 75567, 122502, 198469, 321412, 520365, 842306, 1363247, 2206178, 3570101, 5777008, 9347893, 15125742, 24474535, 39601238, 64076797, 103679124
Offset: 1
Examples
a(5) = a(4)+a(3)+5^2 = 16+37+25 = 78.
Links
- Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).
Formula
a(n) = F(n-2) + (Sum_{i=1..n} i^2) + Sum_{k=0..n-2} F(k)*Sum_{j=0..n-k-1} j^2, where F(i) is the i-th Fibonacci number. [Corrected by Jason Yuen, Apr 09 2025]
G.f.: x*(x^4-4*x^3+6*x^2-3*x+2)/((1-x-x^2)*(1-x)^3). [Corrected by Bruno Berselli, Aug 25 2010 and R. J. Mathar, Oct 18 2010]
Limiting ratio a(n+1)/a(n) = Phi = 1.618038...
a(n) = 2*A022095(n+2)-6*n-13-n^2. - R. J. Mathar, Aug 06 2010
a(n)-4*a(n-1)+5*a(n-2)-a(n-3)-2*a(n-4)+a(n-5) = 0 with n>5. - Bruno Berselli, Aug 25 2010
Comments