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A180312 Number of solutions to n = x + 4*y + 4*z in triangular numbers.

%I A180312 #12 Apr 18 2017 22:43:53
%S A180312 1,1,0,1,2,2,1,2,1,1,3,1,2,2,3,3,2,2,3,4,0,1,4,1,3,5,2,5,3,3,3,4,2,2,
%T A180312 5,0,4,4,2,5,6,2,2,4,5,6,4,2,3,5,4,3,7,3,3,5,2,4,3,4,5,6,2,4,8,6,3,8,
%U A180312 2,4,8,2,6,6,5,4,3,0,5,7,5,5,6,3,5,10,2,6,6,4,10,5,4,3,10,5,4,4,2,9,8,3,7,7,0
%N A180312 Number of solutions to n = x + 4*y + 4*z in triangular numbers.
%C A180312 From page 104 of the Sun reference: "(iii) A positive integer n is a sum of an odd square, an even square and a triangular number, unless it is a triangular number t_m (m>0) for which all prime divisors of 2m+1 are congruent to 1 mod 4 and hence t_m = x^2 + x^2 + t_z for some integers x > 0 and z = x == m/2 (mod 2)."
%C A180312 Numbers of representations of n + 1 as a sum of an odd square, an even square and a triangular number.
%D A180312 Z.-W. Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), no.2, 103--113, see page 104.
%H A180312 G. C. Greubel, <a href="/A180312/b180312.txt">Table of n, a(n) for n = 0..1000</a>
%H A180312 Z.-W. Sun, <a href="http://math.nju.edu.cn/~zwsun/88s.pdf">Mixed sums of squares and triangular numbers</a>
%F A180312 Expansion of q^(-9/8) * eta(q^2)^2 * eta(q^8)^4 / (eta(q) * eta(q^4)^2) in powers of q
%F A180312 Expansion of psi(q) * psi(q^8) * phi(q^4) = psi(q) * psi(q^4)^2 in powers of q where phi(), psi() are Ramanujan theta functions.
%F A180312 Euler transform of period 8 sequence [ 1, -1, 1, 1, 1, -1, 1, -3, ...].
%F A180312 a(n) = 0 if and only if n+1 = A000217(2 * A094178(m)) for some integer m where A000217 is triangular numbers.
%F A180312 G.f.: (Sum_{k>0} x^((n^2 - n)/2)) * (Sum_{k>0} x^(n^2 - n)).
%e A180312 a(10) = 3 since we have 10 = 6 + 4*1 + 4*0 = 6 + 4*0 + 4*1 = 10 + 4*0 + 4*0.
%e A180312 a(10) = 3 since we have 10 + 1 = 1^2 + 0^2 + 10 = 1 + 2^2 + 6 = 1 + (-2)^2 + 6.
%e A180312 1 + x + x^3 + 2*x^4 + 2*x^5 + x^6 + 2*x^7 + x^8 + x^9 + 3*x^10 + x^11 + ...
%t A180312 m=105; psi[q_] = Product[(1-q^(2n))/(1-q^(2n-1)), {n, 1, Floor[m/2]}]; Take[ CoefficientList[ Series[ psi[q]*psi[q^4]^2, {q, 0, m}], q], m] (* _Jean-François Alcover_, Sep 12 2011, after g.f. *)
%o A180312 (PARI) {a(n) = local(A) ; if( n<0, 0, A = x * O(x^n) ; polcoeff( eta(x^2 + A)^2 * eta(x^8 + A)^4 / (eta(x + A) * eta(x^4 + A)^2), n))}
%K A180312 nonn
%O A180312 0,5
%A A180312 _Michael Somos_, Aug 25 2010