A308422 a(n) = n^2 if n odd, 3*n^2/4 if n even.

0, 1, 3, 9, 12, 25, 27, 49, 48, 81, 75, 121, 108, 169, 147, 225, 192, 289, 243, 361, 300, 441, 363, 529, 432, 625, 507, 729, 588, 841, 675, 961, 768, 1089, 867, 1225, 972, 1369, 1083, 1521, 1200, 1681, 1323, 1849, 1452, 2025, 1587, 2209, 1728, 2401, 1875, 2601, 2028, 2809, 2187, 3025

Offset: 0

Author

Ilya Gutkovskiy, May 26 2019

Keywords

• nonn
• easy
• mult

Comments

Moebius transform of A076577.

Links

• Amiram Eldar, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Crossrefs

Cf. A000290, A007434, A016754, A026741, A033428, A076577, A308418, A309337.

Programs

Mathematica

a[n_] := If[OddQ[n], n^2, 3 n^2/4]; Table[a[n], {n, 0, 55}]
nmax = 55; CoefficientList[Series[x (1 + 3 x + 6 x^2 + 3 x^3 + x^4)/(1 - x^2)^3, {x, 0, nmax}], x]
LinearRecurrence[{0, 3, 0, -3, 0, 1}, {0, 1, 3, 9, 12, 25}, 56]
Table[(7 - (-1)^n) n^2/8, {n, 0, 55}]

Formula

G.f.: x*(1 + 3*x + 6*x^2 + 3*x^3 + x^4)/(1 - x^2)^3.

G.f.: Sum_{k>=1} J_2(k)*x^k/(1 - x^(2*k)), where J_2() is the Jordan function (A007434).

E.g.f.: x*((4 + 3*x)*cosh(x) + (3 + 4*x)*sinh(x))/4.

Dirichlet g.f.: zeta(s-2)*(1 - 1/2^s).

a(n) = (7 - (-1)^n)*n^2/8.

a(n) = Sum_{d|n, n/d odd} J_2(d).

a(2*k+1) = A016754(k), a(2*k) = A033428(k).

Sum_{n>=1} 1/a(n) = 13*Pi^2/72 = 1.7820119057522453061...

Sum_{n>=1} (-1)^(n+1)/a(n) = 5*Pi^2/72 = 0.68538919452009434853...

Multiplicative with a(2^e) = 3*2^(2*e-2), and a(p^e) = p^(2*e) for odd primes p. - Amiram Eldar, Oct 26 2020

For n >= 1, n*a(n) = A309337(n) = Sum_{d divides n} (-1)^(d+1) * J(3, n/d), where the Jordan totient function J_3(n) = A059376. - Peter Bala, Jan 21 2024