OFFSET
0,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
FORMULA
Expansion of theta_3(q) * theta_3(q^43) + theta_2(q) * theta_2(q^43) in powers of q.
Expansion of phi(q) * phi(q^43) + 4 * q^11 * psi(q^2) * psi(q^86) in powers of q where phi(), psi() are Ramanujan theta functions.
Moebius transform is period 43 sequence [2, -2, -2, 2, -2, 2, -2, -2, 2, 2, 2, -2, 2, 2, 2, 2, 2, -2, -2, -2, 2, -2, 2, 2, 2, -2, -2, -2, -2, -2, 2, -2, -2, -2, 2, 2, -2, 2, -2, 2, 2, -2, 0, ...].
a(n) = 2*b(n) where b() is multiplicative with b(43^e) = 1, b(p^e) = e + 1 if Kronecker(-43, p) = 1, b(p^e) = (1 + (-1)^e) / 2 otherwise.
G.f. is a period 1 Fourier series which satisfies f(-1 / (43 t)) = 43^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
a(4*n + 2) = a(9*n + 3) = a(9*n + 6) = 0. a(4*n) = a(9*n) = a(n).
G.f.: Sum_{i,j in Z} x^(i*i + i*j + 11*j*j).
a(n) = 2 * A035147(n) unless n = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/sqrt(43) = 0.958176... . - Amiram Eldar, Nov 21 2023
EXAMPLE
G.f. = 1 + 2*q + 2*q^4 + 2*q^9 + 4*q^11 + 4*q^13 + 2*q^16 + 4*q^17 + 4*q^23 + ...
MATHEMATICA
a[ n_] := If[ n < 1, Boole[n == 0], 2 DivisorSum[ n, KroneckerSymbol[ -43, #] &]]; (* Michael Somos, Sep 07 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^43] + EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^43], {q, 0, n}]; (* Michael Somos, Sep 07 2015 *)
Join[{1}, a[n_]:=If[n<0, 0, DivisorSum[n, KroneckerSymbol[-43, #]&]];
2 Table[a[n], {n, 1, 100}]] (* Vincenzo Librandi, Sep 07 2018 *)
PROG
(PARI) {a(n) = if( n<1, n==0, sumdiv(n, d, kronecker(-43, d))*2)};
(PARI) {a(n) = if( n<0, 0, polcoeff( 1 + 2 * x * Ser(qfrep([2, 1; 1, 22], n, 1)), n))};
(PARI) a(n)=if(n, sumdivmult(n, d, kronecker(-43, d))*2, 0) \\ Charles R Greathouse IV, Nov 23 2021
(Magma) A := Basis( ModularForms( Gamma1(43), 1), 87); A[1] + 2*A[2] + 2*A[5] + 2*A[10] + 4*A[12] + 4*A[14] + 2*A[17] + 4*A[18]; /* Michael Somos, Sep 07 2015 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Michael Somos, Mar 31 2008, Apr 05 2008
STATUS
approved