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A254629 Number of ways to write n as x^2 + y*(y+1) + z*(4*z+1) with x,y,z nonnegative integers. 1
1, 1, 1, 1, 1, 1, 3, 2, 1, 2, 1, 3, 2, 1, 1, 2, 3, 1, 3, 1, 3, 5, 2, 1, 3, 3, 2, 3, 2, 3, 3, 3, 1, 2, 4, 1, 5, 1, 2, 5, 2, 3, 5, 4, 1, 4, 4, 3, 4, 4, 2, 5, 2, 1, 4, 5, 5, 3, 1, 1, 7, 5, 1, 3, 4, 2, 5, 3, 2, 6, 5, 3, 4, 4, 5, 5, 4, 4, 5, 3, 1, 8, 2, 4, 7, 3, 4, 3, 5, 3, 6, 3, 3, 3, 6, 4, 5, 5, 2, 9, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

Conjecture: (i) a(n) > 0 for all n, and a(n) > 1 for all n > 338.

(ii) If f(x) is one of the polynomials 3x^2+x, (7x^2+3x)/2, (9x^2-x)/2, (11x^2-7x)/2, (15x^2-7x)/2, (15x^2-11x)/2, then any nonnegative integer n can be written as x^2 + y*(y+1) + f(z) with x,y,z nonnegative integers.

We have proved that for each n = 0,1,... there are integers x,y,z such that n = x^2 + y*(y+1) + z*(4z+1).

It is known that {x^2+y*(y+1): x,y=0,1,...} = {x*(x+1)/2+y*(y+1)/2: x,y=0,1,...}.

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 [math.NT], 2009-2015.

EXAMPLE

a(103) = 1 since 103 = 8^2 + 0*1 + 3*(4*3+1).

a(122) = 1 since 122 = 9^2 + 1*2 +3*(4*3+1).

a(143) = 1 since 143 = 6^2 + 1*2 + 5*(4*5+1).

a(167) = 1 since 167 = 3^2 + 9*10 + 4*(4*4+1).

a(248) = 1 since 248 = 5^2 + 4*5 + 7*(4*7+1).

a(338) = 1 since 338 = 5^2 + 10*11 + 7*(4*7+1).

MATHEMATICA

SQ[n_]:=IntegerQ[Sqrt[n]]

Do[r=0; Do[If[SQ[n-y(y+1)-z*(4z+1)], r=r+1], {y, 0, (Sqrt[4n+1]-1)/2}, {z, 0, (Sqrt[16(n-y(y+1))+1]-1)/8}];

Print[n, " ", r]; Continue, {n, 0, 100}]

CROSSREFS

Cf. A000217, A000290, A007742, A254573, A254574, A254623.

Sequence in context: A342468 A324534 A248505 * A096248 A079109 A079099

Adjacent sequences:  A254626 A254627 A254628 * A254630 A254631 A254632

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Feb 03 2015

STATUS

approved

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Last modified September 20 10:51 EDT 2021. Contains 347584 sequences. (Running on oeis4.)