A275824 Number of ways to write n as pen(x) + pen(y) + pen (z) + 2*pen(w), where x,y,z,w are nonnegative integers with x <= y <= z, and pen(k) denotes the pentagonal number k*(3*k-1)/2.

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

Offset

0,3

Comments

Conjecture: (i) a(n) > 0 for all n = 0,1,2,...., and a(n) = 1 only for n = 0, 1, 4, 6, 8, 9, 18, 19, 21, 33, 40, 43.

(ii) For every m = 5,6,7,... and each integer d > 1 not divisible by 4, any sufficiently large integer can be written as x + y + z + d*w (or x + y + 2*z + d*w) with x,y,z,w m-gonal numbers.

Part (i) of the conjecture is stronger than the classical result that any natural number is the sum of five pentagonal numbers.

Links

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

Xiang-Zi Meng and Zhi-Wei Sun, Sums of four polygonal numbers with coefficients, arXiv:1608.02022 [math.NT], 2016.

Zhi-Wei Sun, A result similar to Lagrange's theorem, J. Number Theory 162(2016), 190-211.

Example

a(4) = 1 since 4 = 0*(3*0-1)/2 + 1*(3*1-1)/2 + 1*(3*1-1)/2 + 1*(3*1-1).
a(18) = 1 since 18 = 1*(3*1-1)/2 + 2*(3*2-1)/2 + 3*(3*3-1)/2 + 0*(3*0-1).
a(19) = 1 since 19 = 0*(3*0-1)/2 + 2*(3*2-1)/2 + 3*(3*3-1)/2 + 1*(3*1-1).
a(21) = 1 since 21 = 1*(3*1-1)/2 + 2*(3*2-1)/2 + 2*(3*2-1)/2 + 2*(3*2-1).
a(33) = 1 since 33 = 0*(3*0-1)/2 + 1*(3*1-1)/2 + 4*(3*4-1)/2 + 2*(3*2-1).
a(40) = 1 since 40 = 0*(3*0-1)/2 + 2*(3*2-1)/2 + 5*(3*5-1)/2 + 0*(3*0-1).
a(43) = 1 since n = 1*(3*1-1)/2 + 2*(3*2-1)/2 + 5*(3*5-1)/2 + 1*(3*1-1).

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
pQ[n_]:=pQ[n]=SQ[24n+1]&&(n==0||Mod[Sqrt[24n+1]+1, 6]==0)
Do[r=0; Do[If[pQ[n-x*(3x-1)-y(3y-1)/2-z(3z-1)/2], r=r+1], {x, 0, (Sqrt[12n+1]+1)/6}, {y, 0, (Sqrt[8(n-x*(3x-1))+1]+1)/6}, {z, y, (Sqrt[12(n-x*(3x-1)-y(3y-1)/2)+1]+1)/6}]; Print[n, " ", r]; Continue, {n, 0, 80}]

Crossrefs

Cf. A000326, A256106.

Sequence in context: A289014 A193238 A323826 * A324869 A167678 A078614

Adjacent sequences: A275821 A275822 A275823 * A275825 A275826 A275827

Keyword

nonn

Author

Zhi-Wei Sun, Aug 10 2016

Status

approved