OFFSET
0,2
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000
Olivia X. M. Yao, Ernest X. W. Xia, Combinatorial proofs of five formulas of Liouville, Discrete Math. 318 (2014), 1--9. MR3141622.
FORMULA
See Maple code.
G.f.: theta_3(q)^2*theta_3(q^4)^2, where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 03 2018
MAPLE
with(numtheory);
s:=n-> if whattype(n) = integer then sigma(n) else 0; fi;
f:=proc(n) global s;
if (n mod 4) = 0 then 8*s(n/4)-32*s(n/16)
elif (n mod 4) = 2 then 4*s(n/2)
elif (n mod 4) = 3 then 0
else 4*s(n); fi; end;
[seq(f(n), n=1..100)];
# a(0)=1 must be added separately
MATHEMATICA
s[n_] := If[IntegerQ[n], DivisorSigma[1, n], 0]; a[n_] := Which[Mod[n, 4] == 0 , 8*s[n/4]-32*s[n/16], Mod[n, 4] == 2, 4*s[n/2], Mod[n, 4] == 3, 0, True, 4*s[n]]; a[0] = 1; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 06 2014, after Maple *)
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Feb 14 2014
STATUS
approved