OFFSET
0,3
COMMENTS
Compare to the g.f. of partitions: Sum_{n>=0} x^n/Product_{k=1..n} (1-x^k).
As an analog to the identity: (1-x^n) = Product_{k=0..n-1} (1 - u^k*x), where u=exp(2*Pi*I/n), we have (1 - A002203(n)*x^n + (-1)^n*x^(2*n)) = Product_{k=0..n-1} (1 - 2*u^k*x - (u^k*x)^2).
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 8*x^3 + 26*x^4 + 67*x^5 + 216*x^6 + 555*x^7 +...
where
A(x) = 1 + x/(1-2*x-x^2) + x^2/((1-2*x-x^2)*(1-6*x^2+x^4)) + x^3/((1-2*x-x^2)*(1-6*x^2+x^4)*(1-14*x^3-x^6)) + x^4/((1-2*x-x^2)*(1-6*x^2+x^4)*(1-14*x^3-x^6)*(1-34*x^4+x^8)) + x^5/((1-2*x-x^2)*(1-6*x^2+x^4)*(1-14*x^3-x^6)*(1-34*x^4+x^8)*(1-82*x^5-x^10)) +...).
The companion Pell numbers begin:
A002203 = [2, 6, 14, 34, 82, 198, 478, 1154, 2786, 6726, 16238, ...].
PROG
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 04 2012
STATUS
approved