OFFSET
1,2
COMMENTS
a(n) = column sums in an array with rows consisting of n' compositions into X and (X+1) parts; n'>0, X=1...inf.
1 1 2 3 5 8 13 21 34 55 89...
1 0 1 1 1 2 2 3 4 5 7
1 0 0 1 1 0 1 2 1 1 3
1 0 0 0 1 1 0 0 1 2 1
1 0 0 0 0 1 1 0 0 0 1
1 0 0 0 0 0 1 1 0 0 0
1 0 0 0 0 0 0 1 1 0 0
1 0 0 0 0 0 0 0 1 1 0
1 0 0 0 0 0 0 0 0 1 1
1 0 0 0 0 0 0 0 0 0 1...
+=1 3 5 8 12 18 28 42 65 103...
- Bob Selcoe, Feb 07 2014
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..1000
FORMULA
G.f.: Sum_{n>=1} x^n * Sum_{d|n} (1+x)^d.
a(n) ~ 1/sqrt(5) * ((1+sqrt(5))/2)^(n+1). - Vaclav Kotesovec, Oct 28 2014
EXAMPLE
G.f.: A(x) = x + 3*x^2 + 5*x^3 + 8*x^4 + 12*x^5 + 18*x^6 + 28*x^7 + 42*x^8 +...
where
A(x) = x*(1+x)/(1-x) + x^2*(1+x)^2/(1-x^2) + x^3*(1+x)^3/(1-x^3) + x^4*(1+x)^4/(1-x^4) + x^5*(1+x)^5/(1-x^5) + x^6*(1+x)^6/(1-x^6) +...
PROG
(PARI) {a(n)=polcoeff(sum(m=1, n, x^m*(1+x)^m/(1-x^m +x*O(x^n)) ), n)}
for(n=1, 40, print1(a(n), ", "))
(PARI) {a(n)=polcoeff(sum(m=1, n, x^m*sumdiv(m, d, (1+x +x*O(x^n))^d) ), n)}
for(n=1, 40, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 18 2013
STATUS
approved