A230121 Number of ways to write n = x + y + z (0 < x <= y <= z) such that x*(x+1)/2 + y*(y+1)/2 + z*(z+1)/2 is a triangular number.

0, 0, 1, 0, 0, 1, 0, 1, 2, 1, 1, 0, 2, 1, 2, 1, 2, 3, 2, 2, 6, 1, 3, 5, 1, 2, 3, 5, 2, 1, 3, 3, 3, 4, 3, 8, 2, 5, 11, 2, 5, 8, 4, 6, 4, 9, 4, 6, 5, 4, 6, 3, 8, 8, 5, 8, 10, 7, 7, 11, 8, 6, 7, 8, 5, 9, 7, 6, 8, 7, 7, 8, 13, 9, 11, 10, 7, 22, 9, 10, 13, 3, 6, 10, 8, 17, 12, 7, 9, 10, 16, 6, 18, 18, 10, 15, 9, 12, 20, 5

Offset

1,9

Comments

Conjecture: (i) a(n) > 0 except for n = 1, 2, 4, 5, 7, 12. Moreover, for each n = 20, 21, ... there are three distinct positive integers x, y and z with x + y + z = n such that x*(x+1)/2 + y*(y+1)/2 + z*(z+1)/2 is a triangular number.

(ii) A positive integer n cannot be written as x + y + z (x, y, z > 0) with x^2 + y^2 + z^2 a square if and only if n has the form 2^r*3^s or the form 2^r*7, where r and s are nonnegative integers.

(iii) Any integer n > 14 can be written as a + b + c + d, where a, b, c, d are positive integers with a^2 + b^2 + c^2 + d^2 a square. If n > 20 is not among 22, 28, 30, 38, 44, 60, then we may require additionally that a, b, c, d are pairwise distinct.

(iv) For each integer n > 50 not equal to 71, there are positive integers a, b, c, d with a + b + c + d = n such that both a^2 + b^2 and c^2 + d^2 are squares.

Part (ii) and the first assertion in part (iii) were confirmed by Chao Huang and Zhi-Wei Sun in 2021. - Zhi-Wei Sun, May 09 2021

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..3000

Chao Huang and Zhi-Wei Sun, On partitions of integers with restrictions involving squares, arXiv:2105.03416 [math.NT], 2021.

Zhi-Wei Sun, Diophantine problems involving triangular numbers and squares, a message to Number Theory List, Oct. 11, 2013.

Example

a(16) = 1 since 16 = 3 + 6 + 7 and 3*4/2 + 6*7/2 + 7*8/2 = 55 = 10*11/2.

Mathematica

SQ[n_]:=IntegerQ[Sqrt[n]]
T[n_]:=n(n+1)/2
a[n_]:=Sum[If[SQ[8(T[i]+T[j]+T[n-i-j])+1], 1, 0], {i, 1, n/3}, {j, i, (n-i)/2}]
Table[a[n], {n, 1, 100}]

Prog

(PARI) a(n)=my(t=(n+1)*n/2, s); sum(x=1, n\3, s=t-n--*x; sum(y=x, n\2, is_A000217(s-(n-y)*y))) \\ - M. F. Hasler, Oct 11 2013

Crossrefs

Cf. A000217, A000290.

Cf. A008443, A053604, A053603, A052343, A052344, A008441.

Sequence in context: A137686 A341973 A143792 * A029375 A071462 A101979

Adjacent sequences: A230118 A230119 A230120 * A230122 A230123 A230124

Keyword

nonn

Author

Zhi-Wei Sun, Oct 10 2013

Status

approved