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%I #18 Jan 21 2024 02:16:23
%S 0,1,6,27,48,125,162,343,384,729,750,1331,1296,2197,2058,3375,3072,
%T 4913,4374,6859,6000,9261,7986,12167,10368,15625,13182,19683,16464,
%U 24389,20250,29791,24576,35937,29478,42875,34992,50653,41154,59319,48000,68921,55566,79507,63888,91125
%N a(n) = n^3 if n odd, 3*n^3/4 if n even.
%C Moebius transform of A078307.
%H Amiram Eldar, <a href="/A309337/b309337.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,4,0,-6,0,4,0,-1).
%F G.f.: x * (1 + 6*x + 23*x^2 + 24*x^3 + 23*x^4 + 6*x^5 + x^6)/(1 - x^2)^4.
%F G.f.: Sum_{k>=1} J_3(k) * x^k/(1 + x^k), where J_3() is the Jordan function (A059376).
%F Dirichlet g.f.: zeta(s-3) * (1 - 2^(1-s)).
%F a(n) = n^3 * (7 - (-1)^n)/8.
%F a(n) = Sum_{d|n} (-1)^(n/d + 1) * J_3(d).
%F Sum_{n>=1} 1/a(n) = 25*zeta(3)/24 = 1.252142607457910713958...
%F 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
%F 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
%t a[n_] := If[OddQ[n], n^3, 3 n^3/4]; Table[a[n], {n, 0, 45}]
%t 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]
%t LinearRecurrence[{0, 4, 0, -6, 0, 4, 0, -1}, {0, 1, 6, 27, 48, 125, 162, 343}, 46]
%t Table[n^3 (7 - (-1)^n)/8, {n, 0, 45}]
%Y Cf. A000578, A016755, A059376, A078307, A129194, A193356, A309338.
%K nonn,easy,mult
%O 0,3
%A _Ilya Gutkovskiy_, Jul 24 2019
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