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A108563 Number of representations of n as sum of twice a square plus thrice a square.

%I A108563 #47 Feb 16 2025 08:32:58
%S A108563 1,0,2,2,0,4,0,0,2,0,0,4,2,0,4,0,0,0,2,0,4,4,0,0,0,0,0,2,0,4,4,0,2,0,
%T A108563 0,8,0,0,0,0,0,0,0,0,4,4,0,0,2,0,6,0,0,4,0,0,4,0,0,4,0,0,4,0,0,0,4,0,
%U A108563 0,0,0,0,2,0,0,6,0,8,0,0,4,0,0,4,4,0,0,0,0,0,0,0,0,4,0,0,0,0,6,4,0,4
%N A108563 Number of representations of n as sum of twice a square plus thrice a square.
%C A108563 Number of solutions to n = 2*a^2 + 3*b^2 in integers.
%C A108563 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A108563 a(n) > 0 if and only if n is in A002480. a(n) < 2 if n is in A002481. - _Michael Somos_, Mar 01 2011
%H A108563 G. C. Greubel, <a href="/A108563/b108563.txt">Table of n, a(n) for n = 0..1000</a>
%H A108563 A. Berkovich and H. Yesilyurt, <a href="https://arxiv.org/abs/math/0611300">Ramanujan's identities and representation of integers by certain binary and quaternary quadratic forms</a>, arXiv:math/0611300 [math.NT], 2006-2007.
%H A108563 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A108563 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A108563 G.f.: 1 + Sum_{k>0} x^k * (1 + x^(4*k)) * (1 + x^(6*k)) / (1 + x^(12*k)) - Sum_{k>0} Kronecker( k, 3) * x^k * (1 - x^(2*k)) / (1 + x^(4*k)).
%F A108563 G.f.: Sum_{i, j in Z} x^(2*i^2 + 3*j^2). - _Michael Somos_, Mar 01 2011
%F A108563 Expansion of phi(q^2) * phi(q^3) in powers of q where phi() is a Ramanujan theta function. - _Michael Somos_, Mar 01 2011
%F A108563 A115660(n) = A000377(n) - a(n). - _Michael Somos_, Mar 01 2011
%F A108563 Euler transform of period 24 sequence [0, 2, 2, -3, 0, -1, 0, -1, 2, 2, 0, -4, 0, 2, 2, -1, 0, -1, 0, -3, 2, 2, 0, -2, ...]. - _Michael Somos_, Jan 20 2017
%F A108563 Expansion of eta(q^4)^5 * eta(q^6)^5 / (eta(q^2)^2 * eta(q^3)^2 * eta(q^8)^2 * eta(q^12)^2) in powers of q. - _Michael Somos_, Jan 20 2017
%F A108563 a(0) = 1, a(n) = (1-(-1)^t)*b(n) for n > 0, where t is the number of prime factors of n, counting multiplicity, which are == 2,3,5,11 (mod 24), and b() is multiplicative with b(p^e) = (e+1) for primes p == 1,5,7,11 (mod 24) and b(p^e) = (1+(-1)^e)/2 for primes p == 13,17,19,23 (mod 24). (This formula is Corollary 4.2 in the Berkovich-Yesilyurt paper). - _Jeremy Lovejoy_, Nov 14 2024
%e A108563 G.f. = 1 + 2*x^2 + 2*x^3 + 4*x^5 + 2*x^8 + 4*x^11 + 2*x^12 + 4*x^14 + 2*x^18 + ...
%e A108563 a(0) = 1 since 0 = 2*0^2 + 3*0^2, a(5) = 4 since 5 = 2*1^2 + 3*1^2 = 2*(-1)^2 + 3*1^2 = 2*1^2 + 3*(-1)^2 = 2*(-1^2) + 3*(-1)^2.
%t A108563 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^2] EllipticTheta[ 3, 0, q^3], {q, 0, n}]; (* _Michael Somos_, Apr 19 2015 *)
%t A108563 a[n_] := Module[{a, b, r}, r = Reduce[n == 2a^2 + 3b^2, {a, b}, Integers]; Which[r === False, 0, r[[0]] === And, 1, r[[0]] === Or, Length[r]]];
%t A108563 Table[a[n], {n, 0, 105}] (* _Jean-François Alcover_, Jan 09 2019 *)
%o A108563 (PARI) for(n=0,120,print1(if(n<1,n==0,qfrep([2,0;0,3],n)[n]*2),","))
%o A108563 (PARI) {a(n) = my(G); if( n<0, 0, G = [2, 0; 0, 3]; polcoeff( 1 + 2 * x * Ser( qfrep( G, n)), n))}; /* _Michael Somos_, Mar 01 2011 */
%o A108563 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^5 * eta(x^6 + A)^5 / (eta(x^2 + A)^2 * eta(x^3 + A)^2 * eta(x^8 + A)^2 * eta(x^12 + A)^2), n))}; /* _Michael Somos_, Jan 20 2017 */
%o A108563 (Sage)
%o A108563 Q = DiagonalQuadraticForm(ZZ,[3, 2])
%o A108563 Q.representation_number_list(102) # _Peter Luschny_, Jun 20 2014
%Y A108563 Cf. A000377, A002480, A002481, A115660.
%K A108563 nonn
%O A108563 0,3
%A A108563 _Ralf Stephan_, May 13 2007
%E A108563 Edited by _Charles R Greathouse IV_, Oct 28 2009
%E A108563 Edited by _N. J. A. Sloane_, Mar 04 2011