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A180312 Number of solutions to n = x + 4*y + 4*z in triangular numbers. 2
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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: A182596 A087775 A089955 * A178819 A046816 A301475

Adjacent sequences:  A180309 A180310 A180311 * A180313 A180314 A180315

KEYWORD

nonn

AUTHOR

Michael Somos, Aug 25 2010

STATUS

approved

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Last modified February 20 22:47 EST 2020. Contains 332086 sequences. (Running on oeis4.)