A145467 Convolution square of A003114.

1, 2, 3, 4, 7, 10, 15, 20, 28, 38, 52, 68, 91, 118, 153, 196, 252, 318, 403, 504, 632, 784, 973, 1196, 1473, 1800, 2198, 2668, 3238, 3908, 4714, 5660, 6789, 8112, 9683, 11516, 13685, 16210, 19178, 22628, 26671, 31354, 36821, 43140, 50489, 58968, 68796

Offset: 0

Author

Michael Somos, Oct 11 2008

Keywords

• nonn

Examples

			1/q + 2*q^29 + 3*q^59 + 4*q^89 + 7*q^119 + 10*q^149 + 15*q^179 + ...
		

Links

• Georg Fischer, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A003114, A145466, A001622.

Programs

Maple

# Using the function EULER from Transforms (see link at the bottom of the page).
[1,op(EULER([seq(op([2,0,0,2,0]),n=1..9)]))]; # Peter Luschny, Aug 19 2020
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*add(`if`(irem(d, 5) in {1, 4}, 2*d, 0),
       d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 19 2020

Mathematica

nmax = 50; CoefficientList[Series[Product[1/((1 - x^(5*k - 1))*(1 - x^(5*k - 4)))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 14 2018 *)

PARI

{a(n) = local(t); if( n<0, 0, t = 1 + x * O(x^n); polcoeff( sum(k=1, sqrtint(n), t *= x^(2*k - 1) / (1 - x^k) * (1 + x * O(x^(n - k^2))), 1)^2, n))}

Formula

a(n) = A145466(5*n).

Expansion of G(x)^2 in powers of x where G() is a Rogers-Ramanujan function.

a(n) ~ exp(2*Pi*sqrt(2*n/15)) * phi / (3^(1/4) * 10^(3/4) * n^(3/4)) * (1 - (3*sqrt(15/2)/(16*Pi) + Pi/(15*sqrt(30)))/sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Aug 14 2018

Euler transform of the period 5 sequence [2, 0, 0, 2, 0, ...]. - Georg Fischer, Aug 19 2020