A037213 Expansion of Sum_{n>=0} n*q^(n^2).

0, 1, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset

0,5

Comments

Multiplicative with a(p^(2e)) = p^e, a(p^(2e+1)) = 0. - Mitch Harris, Jun 09 2005

a(n) is the square root of n if n is square, zero otherwise. - Carl R. White, May 23 2009

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Formula

Dirichlet generating function: zeta(2*s-1). - Franklin T. Adams-Watters, Sep 11 2005.

a(n) = sqrt(n) * floor( cos^2(Pi * sqrt(n)) ). - Carl R. White, May 23 2009

a(n) = A000196(n) * A010052(n). - Reinhard Zumkeller, Jan 27 2010

Sum_{k=1..n} a(k) ~ n/2. - Vaclav Kotesovec, Aug 19 2021

Mathematica

Table[If[IntegerQ[n^(1/2)], n^(1/2), 0], {n, 0, 100}] (* Geoffrey Critzer, Feb 21 2015 *)

Prog

(Haskell)
a037213 n = if n == r ^ 2 then r else 0  where r = a000196 n
a037213_list = zipWith (*) a010052_list a000196_list
-- Reinhard Zumkeller, Nov 09 2012

Crossrefs

Cf. A000196, A010052.

Sequence in context: A112166 A112167 A230571 * A218234 A092197 A213618

Adjacent sequences: A037210 A037211 A037212 * A037214 A037215 A037216

Keyword

nonn,mult

Author

N. J. A. Sloane

Status

approved