A306460 Number of ways to write n as x*(2x-1) + y*(y+1)/2 + z*(z+1)*(z+2)/6, where x,y,z are nonnegative integers with x > 0.

1, 2, 1, 1, 2, 2, 3, 3, 1, 2, 4, 3, 2, 1, 2, 6, 5, 1, 3, 3, 4, 6, 1, 1, 4, 6, 3, 3, 5, 3, 6, 4, 2, 3, 5, 5, 4, 6, 4, 2, 5, 5, 3, 5, 2, 6, 7, 3, 5, 5, 7, 5, 4, 2, 5, 8, 5, 3, 2, 6, 6, 4, 5, 5, 6, 7, 5, 5, 4, 6, 9, 6, 6, 5, 1, 7, 8, 3, 2, 6, 7, 5, 6, 5, 7, 8, 5, 3, 2, 6, 10, 6, 8, 7, 7, 5, 4, 6, 5, 5

Offset

1,2

Comments

Conjecture 1: a(n) > 0 for all n > 0. In other words, any positive integer n can be written as the sum of a positive hexagonal number, a triangular number and a tetrahedral number.

We have verified a(n) > 0 for all n = 1..10^7.

Conjecture 2: Let c be 1 or 3. Then each n = 0,1,... can be written as c*x(x+1) + y*(y+1)/2 + z*(z+1)*(z+2)/6 with x,y,z nonnegative integers.

Conjecture 3: Let t(x) = x*(x+1)*(x+2)/6. Then each n = 0,1,... can be written as 2*t(w) + t(x) + t(y) + t(z) with w,x,y,z nonnegative integers.

We have verified Conjecture 3 for all n = 0..2*10^5. Clearly, Conjecture 3 implies Pollock's conjecture which states that any natural number is the sum of five tetrahedral numbers.

Links

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

Eric Weisstein's World of Mathematics, Tetrahedral Number

Example

a(3) = 1 with 3 = 1*(2*1-1) + 1*2/2 + 1*2*3/6.
a(14) = 1 with 14 = 1*(2*1-1) + 2*3/2 + 3*4*5/6.
a(75) = 1 with 75 = 5*(2*5-1) + 4*5/2 + 4*5*6/2.
a(349) = 1 with 349 = 5*(2*5-1) + 24*25/2 + 2*3*4/6.
a(369) = 1 with 369 = 4*(2*4-1) + 10*11/2 + 11*12*13/6.
a(495) = 1 with 495 = 8*(2*8-1) + 20*21/2 + 9*10*11/6.
a(642) = 1 with 642 = 16*(2*16-1) + 16*17/2 + 3*4*5/6.

Mathematica

f[n_]:=f[n]=n(n+1)(n+2)/6;
TQ[n_]:=TQ[n]=IntegerQ[Sqrt[8n+1]];
tab={}; Do[r=0; Do[If[f[z]>=n, Goto[aa]]; Do[If[TQ[n-f[z]-x(2x-1)], r=r+1], {x, 1, (Sqrt[8(n-f[z])+1]+1)/4}]; Label[aa], {z, 0, n}]; tab=Append[tab, r], {n, 1, 100}]; Print[tab]

Crossrefs

Cf. A000217, A000292, A000384, A000797, A104246, A262813, A306462, A306471.

Sequence in context: A373010 A373111 A351192 * A317805 A231561 A371632

Adjacent sequences: A306457 A306458 A306459 * A306461 A306462 A306463

Keyword

nonn

Author

Zhi-Wei Sun, Feb 17 2019

Status

approved