A307981 Number of ways to write n as x^3 + 2*y^3 + 3*z^3 + w*(w+1)*(w+2)/6, where x,y,z,w are nonnegative integers.

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

Offset

0,2

Comments

Conjecture: a(n) > 0 for every nonnegative integer n. In other words, we have {x^3 + 2*y^3 + 3*z^3 + w*(w+1)*(w+2)/6: x,y,z,w = 0,1,2,...} = {0,1,2,...}.

We have verified a(n) > 0 for all n = 0..2*10^6.

Links

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

Example

a(19) = 1 with 19 = 0^3 + 2*2^3 + 3*1^3 + 0*1*2/6.
a(22) = 1 with 22 = 0^3 + 2*1^3 + 3*0^3 + 4*5*6/6.
a(112) = 1 with 112 = 3^3 + 2*0^3 + 3*3^3 + 2*3*4/6.
a(158) = 1 with 158 = 3^3 + 2*4^3 + 3*1^3 + 0*1*2/6.
a(791) = 1 with 791 = 1^3 + 2*5^3 + 3*5^3 + 9*10*11/6.
a(956) = 1 with 956 = 9^3 + 2*0^3 + 3*4^3 + 5*6*7/6.
a(6363) = 1 with 6363 = 10^3 + 2*13^3 + 3*0^3 + 17*18*19/6.

Mathematica

CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)]; f[w_]:=f[w]=Binomial[w+2, 3];
tab={}; Do[r=0; w=0; Label[bb]; If[f[w]>n, Goto[aa]]; Do[If[CQ[n-f[w]-2y^3-3z^3], r=r+1], {y, 0, ((n-f[w])/2)^(1/3)}, {z, 0, ((n-f[w]-2y^3)/3)^(1/3)}]; w=w+1; Goto[bb]; Label[aa]; tab=Append[tab, r], {n, 0, 100}]; Print[tab]

Crossrefs

Cf. A000292, A000578, A262813, A306460, A306477, A306790.

Sequence in context: A144909 A352636 A117114 * A262824 A242899 A347865

Adjacent sequences: A307978 A307979 A307980 * A307982 A307983 A307984

Keyword

nonn,look

Author

Zhi-Wei Sun, May 08 2019

Status

approved