A306790 Number of ways to write n as x*(3*x+1)/2 + y*(y+1)*(y+2)/2 + z*(z+1)*(z+2)/6, where x is a nonzero integer, and y and z are nonnegative integers.

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

Offset

1,2

Comments

Conjecture: a(n) > 0 for all n > 0. In other words, each positive integer n can be written as the sum of a positive generalized pentagonal number, a tetrahedral number and a tetrahedral number times three.

This has been verified for all n = 1..2*10^7.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Example

a(59) = 1 with 59 = (-3)*(3*(-3)+1)/2 + 2*3*4/2 + 5*6*7/6.
a(19694) = 1 with 19694 = 20*(3*20+1)/2 + 10*11*12/2 + 47*48*49/6.
a(19919) = 1 with 19919 = (-45)*(3*(-45)+1)/2 + 30*31*32/2 + 22*23*24/6.
a(33989) = 1 with 33989 = 55*(3*55+1)/2 + 20*21*22/2 + 52*53*54/6.
a(60769) = 1 with 60769 = 46*(3*46+1)/2 + 47*48*49/2 + 23*24*25/6.

Mathematica

f[n_]:=f[n]=Binomial[n+2, 3]; PQ[n_]:=PQ[n]=IntegerQ[Sqrt[24n+1]];
tab={}; Do[r=0; Do[If[f[x]>=n/3, Goto[cc]]; Do[If[f[y]>=n-3*f[x], Goto[bb]]; If[PQ[n-3*f[x]-f[y]], r=r+1]; Label[aa], {y, 0, n-1-3*f[x]}]; Label[bb], {x, 0, (n-1)/3}]; Label[cc]; tab=Append[tab, r], {n, 1, 100}]; Print[tab]

Crossrefs

Cf. A000292, A001318, A306460, A306477.

Sequence in context: A302654 A264831 A264728 * A332056 A074744 A341474

Adjacent sequences: A306787 A306788 A306789 * A306791 A306792 A306793

Keyword

nonn

Author

Zhi-Wei Sun, Mar 10 2019

Status

approved