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A050471 a(n) = Sum_{d|n, n/d=1 mod 4} d^3 - Sum_{d|n, n/d=3 mod 4} d^3.

%I A050471 #36 Sep 26 2024 16:20:57
%S A050471 1,8,26,64,126,208,342,512,703,1008,1330,1664,2198,2736,3276,4096,
%T A050471 4914,5624,6858,8064,8892,10640,12166,13312,15751,17584,18980,21888,
%U A050471 24390,26208,29790,32768,34580,39312,43092,44992,50654,54864,57148
%N A050471 a(n) = Sum_{d|n, n/d=1 mod 4} d^3 - Sum_{d|n, n/d=3 mod 4} d^3.
%C A050471 Multiplicative because it is the Dirichlet convolution of A000578 = n^3 and A101455 = [1 0 -1 0 1 0 -1 ...], which are both multiplicative. - _Christian G. Bower_, May 17 2005
%H A050471 Seiichi Manyama, <a href="/A050471/b050471.txt">Table of n, a(n) for n = 1..10000</a>
%H A050471 J. W. L. Glaisher, <a href="https://books.google.com/books?id=bLs9AQAAMAAJ&amp;pg=RA1-PA1">On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares</a>, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
%H A050471 <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>.
%F A050471 G.f.: Sum_{n>=1} n^3*x^n/(1+x^(2*n)). - _Vladeta Jovovic_, Oct 16 2002
%F A050471 From _Amiram Eldar_, Nov 04 2023: (Start)
%F A050471 Multiplicative with a(p^e) = (p^(3*e+3) - A101455(p)^(e+1))/(p^3 - A101455(p)).
%F A050471 Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = A175572. (End)
%F A050471 a(n) = Sum_{d|n} (n/d)^3*sin(d*Pi/2). - _Ridouane Oudra_, Sep 26 2024
%t A050471 max = 40; s = Sum[n^3*x^(n-1)/(1+x^(2*n)), {n, 1, max}] + O[x]^max; CoefficientList[s, x] (* _Jean-François Alcover_, Dec 02 2015, after _Vladeta Jovovic_ *)
%t A050471 s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
%t A050471 f[p_, e_] := (p^(3*e+3) - s[p]^(e+1))/(p^3 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 04 2023 *)
%o A050471 (PARI) a(n) = sumdiv(n, d, d^3*(((n/d) % 4)==1)) - sumdiv(n, d, d^3*(((n/d) % 4)==3)); \\ _Michel Marcus_, Feb 16 2015
%Y A050471 Cf. A000578, A101455, A175572.
%Y A050471 Glaisher's E'_i (i=0..12): A002654, A050469, A050470, this sequence, A050468, A321829, A321830, A321831, A321832, A321833, A321834, A321835, A321836.
%K A050471 nonn,easy,mult
%O A050471 1,2
%A A050471 _N. J. A. Sloane_, Dec 23 1999
%E A050471 Offset changed from 0 to 1 by _R. J. Mathar_, Jul 15 2010