

A160324


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


10



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
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OFFSET

0,2


COMMENTS

In April 2009, ZhiWei 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, ZhiWei 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)=(k2)x(x1)/2+x (x=0,1,2,...) are kgonal 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.  ZhiWei Sun, Aug 15 2009
On Aug 21 2009, ZhiWei 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,...,m3. 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).  ZhiWei Sun, Aug 21 2009
On Sep 04 2009, ZhiWei 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.  ZhiWei Sun, Sep 04 2009


LINKS

ZhiWei 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), 2224.
G. Pall, Large positive integers are sums of four or five values of a quadratic function, Amer. J. Math. 54(1932), 6678.
ZhiWei Sun, Various new conjectures involving polygonal numbers and primes (a message to Number Theory List), May 2009.
ZhiWei Sun, Mixed Sums of Primes and Other Terms (a webpage).
ZhiWei Sun, On universal sums of polygonal numbers, preprint, arXiv:0905.0635 [math.NT], 2009.


FORMULA

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


EXAMPLE

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


MATHEMATICA

SQ[x_]:=x>1&&IntegerPart[Sqrt[x]]^2==x RN[n_]:=Sum[If[SQ[n(3y^2y)/2(2z^2z)], 1, 0], {y, 0, Sqrt[n]}, {z, 0, Sqrt[Max[0, n(3y^2y)/2]]}] Do[Print[n, " ", RN[n]], {n, 0, 50000}]
planeFiguratePi[n_, r_] := Floor[((r 4) + Sqrt[(r 4)^2 + 8n (r 2)])/(2 (r 2))]; z[r_] := PolygonalNumber[r, Range[0, planeFiguratePi[mx, r]]]; mx = 105; Join[{1}, Take[Transpose[ Tally[ Sort[ Plus @@@ FlattenAt[ Tuples[{z[4], z[5], z[6]}], 2]]]][[2]], {2, mx}]] (* Robert G. Wilson v, May 22 2017 *)


CROSSREFS

Cf. A000290, A000326, A000384.
Sequence in context: A266509 A266539 A090569 * A197928 A109439 A247646
Adjacent sequences: A160321 A160322 A160323 * A160325 A160326 A160327


KEYWORD

nonn


AUTHOR

ZhiWei Sun, May 08 2009


STATUS

approved



