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

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, 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, 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

Offset: 0

Author

Michael Somos, Aug 25 2010

Keywords

• nonn

Comments

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)."

Numbers of representations of n + 1 as a sum of an odd square, an even square and a triangular number.

Examples

			a(10) = 3 since we have 10 = 6 + 4*1 + 4*0 = 6 + 4*0 + 4*1 = 10 + 4*0 + 4*0.
a(10) = 3 since we have 10 + 1 = 1^2 + 0^2 + 10 = 1 + 2^2 + 6 = 1 + (-2)^2 + 6.
1 + x + x^3 + 2*x^4 + 2*x^5 + x^6 + 2*x^7 + x^8 + x^9 + 3*x^10 + x^11 + ...
		

References

• Z.-W. Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), no.2, 103--113, see page 104.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Z.-W. Sun, Mixed sums of squares and triangular numbers

Programs

Mathematica

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. *)

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))}

Formula

Expansion of q^(-9/8) * eta(q^2)^2 * eta(q^8)^4 / (eta(q) * eta(q^4)^2) in powers of q

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.

Euler transform of period 8 sequence [ 1, -1, 1, 1, 1, -1, 1, -3, ...].

a(n) = 0 if and only if n+1 = A000217(2 * A094178(m)) for some integer m where A000217 is triangular numbers.

G.f.: (Sum_{k>0} x^((n^2 - n)/2)) * (Sum_{k>0} x^(n^2 - n)).