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

0,5

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.

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

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)).

Example

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 + ...

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. *)

Prog

(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))}

Crossrefs

Sequence in context: A366091 A089955 A352942 * A178819 A369174 A355855

Adjacent sequences: A180309 A180310 A180311 * A180313 A180314 A180315

Keyword

nonn

Author

Michael Somos, Aug 25 2010

Status

approved