OFFSET
0,3
COMMENTS
Moebius transform of A078307.
LINKS
Amiram Eldar, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-6,0,4,0,-1).
FORMULA
G.f.: x * (1 + 6*x + 23*x^2 + 24*x^3 + 23*x^4 + 6*x^5 + x^6)/(1 - x^2)^4.
G.f.: Sum_{k>=1} J_3(k) * x^k/(1 + x^k), where J_3() is the Jordan function (A059376).
Dirichlet g.f.: zeta(s-3) * (1 - 2^(1-s)).
a(n) = n^3 * (7 - (-1)^n)/8.
a(n) = Sum_{d|n} (-1)^(n/d + 1) * J_3(d).
Sum_{n>=1} 1/a(n) = 25*zeta(3)/24 = 1.252142607457910713958...
Multiplicative with a(2^e) = 3*2^(3*e-2), and a(p^e) = p^(3*e) for odd primes p. - Amiram Eldar, Oct 26 2020
a(n) = Sum_{1 <= i, j, k <= n} (-1)^(1 + gcd(i,j,k,n)) = Sum_{d | n} (-1)^(d+1) * J_3(n/d). Cf. A129194. - Peter Bala, Jan 16 2024
MATHEMATICA
a[n_] := If[OddQ[n], n^3, 3 n^3/4]; Table[a[n], {n, 0, 45}]
nmax = 45; CoefficientList[Series[x (1 + 6 x + 23 x^2 + 24 x^3 + 23 x^4 + 6 x^5 + x^6)/(1 - x^2)^4, {x, 0, nmax}], x]
LinearRecurrence[{0, 4, 0, -6, 0, 4, 0, -1}, {0, 1, 6, 27, 48, 125, 162, 343}, 46]
Table[n^3 (7 - (-1)^n)/8, {n, 0, 45}]
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Ilya Gutkovskiy, Jul 24 2019
STATUS
approved