This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A045833 #29 Nov 14 2024 17:08:46
%S A045833 0,1,0,0,1,0,0,2,0,0,0,0,0,2,0,0,1,0,0,2,0,0,0,0,0,1,0,0,2,0,0,2,0,0,
%T A045833 0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,3,0,0,2,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,
%U A045833 0,0,0,0,0,2,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,2,0,0,1,0,0,2,0
%N A045833 Expansion of eta(q^9)^3 / eta(q^3) in powers of q.
%H A045833 Amiram Eldar, <a href="/A045833/b045833.txt">Table of n, a(n) for n = 0..10000</a>
%F A045833 From _Michael Somos_, May 25 2005: (Start)
%F A045833 Euler transform of period 9 sequence [ 0, 0, 1, 0, 0, 1, 0, 0, -2, ...].
%F A045833 G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2*w - 2*u*w^2 - v^3.
%F A045833 G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1*u3^2 + u1*u6^2 - u1*u3*u6 - u2^2*u3.
%F A045833 a(3*n) = a(3*n + 2) = 0. a(3*n + 1) = A033687(n). a(6*n + 1) = A097195(n). 3*a(n) = A033685(n).
%F A045833 Multiplicative with a(3^e) = 0^e, a(p^e) = e+1 if p == 1 (mod 3), a(p^e) = (1+(-1)^e)/2 if p == 2 (mod 3).
%F A045833 G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u2*u3^2 + 2*u2*u3*u6 + 4*u2*u6^2 - u1^2*u6. (End)
%F A045833 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/(9*sqrt(3)) = 0.403066... . - _Amiram Eldar_, Oct 13 2022
%F A045833 Dirichlet g.f.: L(chi_1,s)*L(chi_{-1},s), where chi_1 = A011655 and chi_{-1} = A102283 are respectively the principal and the non-principal Dirichlet character modulo 3. For the formula of the sequence whose Dirichlet g.f. is Product_{chi} L(chi,s), where chi runs through all Dirichlet characters modulo k, see A378006. This sequence is the case k = 3. - _Jianing Song_, Nov 13 2024
%e A045833 G.f. = q + q^4 + 2*q^7 + 2*q^13 + q^16 + 2*q^19 + q^25 + 2*q^28 + 2*q^31 + ...
%t A045833 a[ n_] := SeriesCoefficient[ q QPochhammer[ q^9]^3 / QPochhammer[ q^3], {q, 0, n}]; (* _Michael Somos_, Feb 22 2015 *)
%t A045833 f[p_, e_] := If[Mod[p, 3] == 1, e + 1, (1 + (-1)^e)/2]; f[3, e_] := 0; a[0] = 0; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100, 0] (* _Amiram Eldar_, Oct 13 2022 *)
%o A045833 (PARI) {a(n) = local(A, p, e); if( n<0, 0, A=factor(n); prod(k=1, matsize(A)[1], if( p=A[k,1], e=A[k,2]; if( p!=3, if( p%3==1, e+1, !(e%2))))))}; \\ _Michael Somos_, May 25 2005
%o A045833 (PARI) {a(n) = local(A); if( (n<1) || (n%3!=1), 0, n = (n-1)/3; A = x * O(x^n); polcoeff( eta(x^3 + A)^3 / eta(x + A), n))}; \\ _Michael Somos_, May 25 2005
%Y A045833 Cf. A033685, A033687, A097195, A217219.
%K A045833 nonn,mult
%O A045833 0,8
%A A045833 _N. J. A. Sloane_