A350143 a(n) = Sum_{k=1..n} floor(n/(2*k-1))^2.

1, 4, 10, 17, 27, 41, 55, 70, 93, 115, 137, 167, 193, 223, 267, 298, 332, 381, 419, 465, 525, 571, 617, 679, 738, 792, 868, 930, 988, 1080, 1142, 1205, 1297, 1367, 1459, 1560, 1634, 1712, 1820, 1914, 1996, 2120, 2206, 2300, 2450, 2544, 2638, 2764, 2875, 2996, 3136, 3246

Offset

1,2

Links

Table of n, a(n) for n=1..52.

Formula

a(n) = Sum_{k=1..n} Sum_{d|k, k/d odd} 2*d - 1 = Sum_{k=1..n} 2 * A002131(k) - A001227(k) = 2 * A350146(n) - A060831(n).

G.f.: (1/(1 - x)) * Sum_{k>=1} (2*k - 1) * x^k/(1 - x^(2*k)).

Mathematica

a[n_] := Sum[Floor[n/(2*k - 1)]^2, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Dec 17 2021 *)

Prog

(PARI) a(n) = sum(k=1, n, (n\(2*k-1))^2);
(PARI) a(n) = sum(k=1, n, sumdiv(k, d, k/d%2*(2*d-1)));
(PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, (2*k-1)*x^k/(1-x^(2*k)))/(1-x))

Crossrefs

Column 2 of A350122.

Cf. A001227, A002131, A060831, A350146.

Sequence in context: A138105 A213398 A002442 * A301253 A341772 A301195

Adjacent sequences: A350140 A350141 A350142 * A350144 A350145 A350146

Keyword

nonn

Author

Seiichi Manyama, Dec 16 2021

Status

approved