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

0,2

Links

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

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

Example

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

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 *)

Prog

(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))}

Crossrefs

Cf. A003114, A145466, A001622.

Sequence in context: A347734 A267459 A330357 * A325341 A039841 A078159

Adjacent sequences: A145464 A145465 A145466 * A145468 A145469 A145470

Keyword

nonn

Author

Michael Somos, Oct 11 2008

Status

approved