A254573 Number of ways to write n = x*(x+1) + y*(3*y+1)/2 + z*(3*z-1)/2 with x,y,z nonnegative integers

1, 1, 2, 2, 1, 2, 1, 4, 2, 3, 1, 1, 3, 3, 5, 2, 2, 2, 3, 3, 4, 3, 4, 1, 4, 2, 4, 5, 4, 3, 2, 4, 5, 4, 2, 4, 2, 6, 3, 5, 3, 3, 6, 5, 5, 3, 3, 6, 2, 6, 5, 3, 4, 3, 6, 2, 4, 9, 6, 4, 4, 5, 5, 5, 7, 3, 2, 3, 8, 4, 6

Offset

0,3

Comments

Conjecture: a(n) > 0 for all n. Also, a(n) = 1 only for n = 0, 1, 4, 6, 10, 11, 23.

This has been verified for all n = 0..10^7. We have proved that every nonnegative integer can be written as x*(x+1) + y*(3*y+1)/2 + z*(3*z-1)/2 with x,y,z integers.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000

Zhi-Wei Sun, On universal sums of polygonal numbers, arXiv:0905.0635.

Example

a(10) = 1 since 10 = 1*2 + 2*(3*2+1)/2 + 1*(3*1-1)/2.
a(11) = 1 since 11 = 2*3 + 0*(3*0+1)/2 + 2*(3*2-1)/2.
a(23) = 1 since 23 = 4*5 + 1*(3*1+1)/2 + 1*(3*1-1)/2.
a(34) = 2 since 34 = 3*4 + 0*(3*0+1)/2 + 4*(3*4-1)/2 = 4*5 + 1*(3*1+1)/2 + 3*(3*3-1)/2.

Mathematica

sQ[n_]:=IntegerQ[Sqrt[4n+1]]
Do[r=0; Do[If[sQ[n-y(3y+1)/2-z(3z-1)/2], r=r+1], {y, 0, (Sqrt[24n+1]-1)/6}, {z, 0, (Sqrt[24(n-y(3y+1)/2)+1]+1)/6}];
Print[n, " ", r]; Continue, {n, 0, 70}]

Crossrefs

Cf. A000326, A002378, A005449, A254574.

Sequence in context: A343491 A326952 A109909 * A144387 A030768 A051480

Adjacent sequences: A254570 A254571 A254572 * A254574 A254575 A254576

Keyword

nonn

Author

Zhi-Wei Sun, Feb 01 2015

Status

approved