A270616 Number of ordered ways to write n as the sum of a positive square, the square of a triangular number, and a generalized pentagonal number (A001318).

1, 2, 2, 2, 2, 3, 2, 1, 3, 4, 4, 3, 2, 3, 3, 4, 6, 4, 3, 3, 2, 3, 3, 3, 6, 4, 5, 4, 1, 4, 4, 5, 2, 1, 3, 5, 6, 5, 6, 5, 5, 5, 2, 5, 6, 3, 5, 3, 5, 6, 6, 10, 4, 2, 3, 4, 5, 4, 5, 7, 6, 5, 4, 4, 6, 6, 7, 2, 3, 3, 6, 6, 5, 6, 5, 6, 5, 3, 4, 8

Offset: 1

Author

Zhi-Wei Sun, Mar 20 2016

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 8, 29, 34, 5949, 10913.

See also A270566 and A270594 for more similar conjectures.

By the author's work in Sci. China Math. 58(2015), any natural number can be written as the sum of a triangular number, a square and a generalized pentagonal number.

Examples

			a(1) = 1 since 1 = 1^2 + (0*1/2)^2 + 0*(3*0+1)/2.
a(8) = 1 since 8 = 1^2 + (0*1/2)^2 + 2*(3*2+1)/2.
a(29) = 1 since 29 = 4^2 + (1*2/2)^2 + 3*(3*3-1)/2.
a(34) = 1 since 34 = 5^2 + (2*3/2)^2 + 0*(3*0+1)/2.
a(5949) = 1 since 5949 = 47^2 + (10*11/2)^2 + 22*(3*22-1)/2.
a(10913) = 1 since 10913 = 23^2 + (2*3/2)^2 +83*(3*83+1)/2.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127 (2007), 103-113.
• Zhi-Wei Sun, On universal sums of polygonal numbers, Sci. China Math. 58 (2015), no. 7, 1367-1396.
• Zhi-Wei Sun, On universal sums ax^2+by^2+f(z), aT_x+bT_y+f(z) and zT_x+by^2+f(z), preprint, arXiv:1502.03056 [math.NT], 2015.

Crossrefs

Cf. A000217, A000290, A001318, A262813, A262815, A262816, A262941, A262944, A262945, A262954, A262955, A262956, A270469, A270488, A270516, A270533, A270559, A270566, A270594.

Programs

Mathematica

pQ[n_]:=pQ[n]=IntegerQ[Sqrt[24n+1]]
Do[r=0;Do[If[pQ[n-x^2-(y(y+1)/2)^2],r=r+1],{x,1,Sqrt[n]},{y,0,(Sqrt[8*Sqrt[n-x^2]+1]-1)/2}];Print[n," ",r];Continue,{n,1,80}]