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A306239
Number of ways to write n as x^3 + y^3 + pen(z) + pen(w), where x, y, z, w are nonnegative integers with x <= y and z <= w, and pen(k) denotes the pentagonal number k*(3*k-1)/2.
4
1, 2, 3, 2, 1, 1, 2, 2, 2, 2, 3, 2, 2, 3, 4, 2, 1, 2, 3, 2, 1, 3, 3, 2, 3, 3, 3, 2, 4, 4, 2, 2, 3, 4, 2, 4, 5, 4, 3, 2, 5, 3, 2, 3, 4, 4, 1, 2, 3, 3, 3, 4, 6, 3, 3, 3, 4, 3, 3, 4, 4, 2, 3, 4, 5, 5, 4, 4, 2, 2, 5, 8, 7, 4, 4, 5, 3, 5, 6, 7, 2, 3, 5, 3, 5, 2, 5, 5, 4, 4, 3, 6, 5, 4, 6, 3, 2, 4, 8, 5, 5
OFFSET
0,2
COMMENTS
Conjecture: a(n) > 0 for all n >= 0, and a(n) = 1 only for n = 0, 4, 5, 16, 20, 46. Also, any nonnegative integer not equal to 16 can be written as x^6 + y^3 + pen(z) + pen(w) with x, y, z, w nonnegative integers.
We have verified a(n) > 0 for all n = 0..5*10^6.
EXAMPLE
a(4) = 1 with 4 = 1^3 + 1^3 + pen(1) + pen(1).
a(5) = 1 with 5 = 0^3 + 0^3 + pen(0) + pen(2).
a(16) = 1 with 16 = 2^3 + 2^3 + pen(0) + pen(0).
a(20) = 1 with 20 = 0^3 + 2^3 + pen(0) + pen(3).
a(46) = 1 with 46 = 1^3 + 1^3 + pen(4) + pen(4).
MATHEMATICA
PenQ[n_]:=PenQ[n]=IntegerQ[Sqrt[24n+1]]&&(n==0||Mod[Sqrt[24n+1]+1, 6]==0);
tab={}; Do[r=0; Do[If[PenQ[n-x^3-y^3-z(3z-1)/2], r=r+1], {x, 0, (n/2)^(1/3)}, {y, x, (n-x^3)^(1/3)}, {z, 0, (Sqrt[12(n-x^3-y^3)+1]+1)/6}]; tab=Append[tab, r], {n, 0, 100}]; Print[tab]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jan 31 2019
STATUS
approved