A160324 Number of ways to express n as the sum of a square, a pentagonal number and a hexagonal number.

1, 3, 3, 1, 1, 3, 4, 3, 1, 2, 4, 3, 2, 2, 2, 4, 5, 4, 2, 2, 3, 3, 5, 3, 3, 2, 3, 5, 4, 5, 2, 5, 5, 2, 2, 1, 6, 8, 5, 2, 3, 5, 4, 3, 4, 5, 3, 3, 2, 5, 7, 7, 5, 4, 7, 4, 4, 3, 4, 4, 3, 6, 3, 2, 5, 5, 9, 7, 3, 3, 6, 9, 5, 3, 1, 8, 7, 6, 2, 5, 6, 3, 10, 4, 3, 3, 8, 7, 5, 4, 1, 4, 10, 7, 5, 4, 8, 6, 2, 8, 6, 10, 7, 5

Offset: 0

Author

Zhi-Wei Sun, May 08 2009

Keywords

• nonn

Comments

In April 2009, Zhi-Wei Sun conjectured that a(n)>0 for every n=0,1,2,3,.... Note that pentagonal numbers and hexagonal numbers are more sparse than squares and that there are infinitely many positive integers which cannot be written as the sum of three squares.

On Aug 12 2009, Zhi-Wei Sun made the following general conjecture on diagonal representations by polygonal numbers: For each integer m>2, any natural number n can be written in the form p_{m+1}(x_1)+...+p_{2m}(x_m) with x_1,...,x_m nonnegative integers, where p_k(x)=(k-2)x(x-1)/2+x (x=0,1,2,...) are k-gonal numbers. Sun has verified this with m=3 for n up to 10^6, and with m=4,5,6,7,8,9,10 for n up to 5*10^5. - Zhi-Wei Sun, Aug 15 2009

On Aug 21 2009, Zhi-Wei Sun formulated the following strong version for his conjecture on diagonal representations by polygonal numbers: For any integer m>2, each natural number n can be expressed as p_{m+1}(x_1)+p_{m+2}(x_2)+p_{m+3}(x_3)+r with x_1,x_2,x_3 nonnegative integers and r an integer among 0,...,m-3. For m=3 and m=4,5,6,7,8,9,10, Sun has verified this conjecture for n up to 10^6 and 5*10^5 respectively. Sun also guessed that for each m=3,4,... all sufficiently large integers have the form p_{m+1}(x_1)+p_{m+2}(x_2)+p_{m+3}(x_3) with x_1,x_2,x_3 nonnegative integers. For example, it seems that 387904 is the largest integer not in the form p_{20}(x_1)+p_{21}(x_2)+p_{22}(x_3). - Zhi-Wei Sun, Aug 21 2009

On Sep 04 2009, Zhi-Wei Sun conjectured that the sequence contains every positive integer. For n=1,2,3,... let s(n) denote the least nonnegative integer m such that a(m)=n. Here is the list of s(1),...,s(30): 0, 9, 1, 6, 16, 36, 50, 37, 66, 82, 167, 121, 162, 236, 226, 276, 302, 446, 478, 532, 457, 586, 677, 521, 666, 852, 976, 877, 1006, 1046. - Zhi-Wei Sun, Sep 04 2009

Examples

			For n=10 the a(10)=4 solutions are 4+0+6, 4+5+1, 9+0+1, 9+1+0.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 0..50000
• M. B. Nathanson, A short proof of Cauchy's polygonal number theorem, Proc. Amer. Math. Soc. 99(1987), 22-24.
• G. Pall, Large positive integers are sums of four or five values of a quadratic function, Amer. J. Math. 54(1932), 66-78.
• Zhi-Wei Sun, Various new conjectures involving polygonal numbers and primes (a message to Number Theory List), May 2009.
• Zhi-Wei Sun, Mixed Sums of Primes and Other Terms (a webpage).
• Zhi-Wei Sun, On universal sums of polygonal numbers, arXiv:0905.0635 [math.NT], 2009-2015.

Crossrefs

Cf. A000290, A000326, A000384, A008443, A240088.

Programs

Mathematica

a = Compile[{{n, Integer}}, Block[{c = 0, h, j, k = Floor[ (Sqrt[ 8n +1] +1)/4]}, While[k > -1, h = k( 2k -1); j = Floor[ (Sqrt[ 24( n -h) +1] +1)/6]; While[j > -1, c += Boole[ Mod[ Sqrt[ n -h -j( 3j -1)/2], 1] == 0]; j--]; k--]; c]]; Array[a, 111, 0] (* _Robert G. Wilson v, Aug 06 2025 *)

Formula

a(n) = |{: x,y,z=0,1,2,... & x^2+(3y^2-y)/2+(2z^2-z)=n}|.