A306382 -id:A306382 - cp's OEIS frontend

A306383 Number of ways to write n as x(2x+1) + y(2y+1) + z*(2z+1), where x,y,z are nonnegative integers with x <= y <= z.

1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 2, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 2, 1, 0, 2, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 2, 0, 1, 1, 0, 0, 1, 1, 1, 0

Offset: 0

Zhi-Wei Sun, Feb 11 2019

nonn

Conjecture 1: a(n) > 0 for any integer n > 138158.
We have verified this for n up to 2*10^6. Note that n*(2n+1) (n = 0,1,...) are the second hexagonal numbers (A014105).
Conjecture 2: Any integer n > 146858 can be written as the sum of three hexagonal numbers (A000384).
Conjecture 3: Any integer n > 33066 can be written as the sum of three pentagonal numbers (A000326).
Conjecture 4: Any integer n > 24036 can be written as the sum of three second pentagonal numbers (A005449).
Conjecture 5: Let N(1) = 114862, N(-1) = 166897, N(3) = 196987 and N(-3) = 273118. Then, for any r among 1, -1, 3 and -3, each integer n > N(r) can be written as x*(5x+r)/2 + y*(5y+r)/2 + z*(5z+r)/2 with x,y,z nonnegative integers.
We have verified Conjectures 2-5 for n up to 10^6.

			a(223595) = 1 with 223595 = 95*(2*95+1) + 200*(2*200+1) + 250*(2*250+1).
a(290660) = 1 with 290660 = 136*(2*136+1) + 149*(2*149+1) + 323*(2*323+1).

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, Universal sums of three quadratic polynomials, Sci. China Math., in press.

Cf. A000217, A000326, A000384, A000566, A005449, A005475, A005476, A008443, A014105, A147875, A306382.

QQ[n_]:=QQ[n]=IntegerQ[Sqrt[8n+1]]&&Mod[Sqrt[8n+1],4]==1;
tab={};Do[r=0;Do[If[QQ[n-x(2x+1)-y(2y+1)],r=r+1],{x,0,(Sqrt[8n/3+1]-1)/4},{y,x,(Sqrt[4(n-x(2x+1))+1]-1)/4}];tab=Append[tab,r],{n,0,100}];Print[tab]

A306439 Number of ways to write n as x(3x+1)/2 + y(3y+1)/2 + z(3z+1) + 3w(3w+1)/2, where x,y,z,w are nonnegative integers with x <= y.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 2, 1, 2, 1, 2, 1, 1, 2, 3, 2, 1, 2, 2, 2, 2, 4, 2, 3, 2, 3, 2, 3, 4, 3, 4, 1, 5, 1, 5, 3, 5, 4, 3, 4, 5, 1, 5, 3, 4, 4, 3, 7, 2, 4, 4, 7, 6, 6, 4, 4, 5, 3, 7, 5, 5, 8, 6, 7, 3, 6, 8, 6, 5, 4, 3, 4, 6, 7, 3, 7, 6, 10, 7, 5, 9, 3, 11, 4, 9, 7, 7, 10, 5, 9, 7, 7, 10, 8, 7, 5, 5, 9, 5, 9, 9

Offset: 0

Zhi-Wei Sun, Feb 15 2019

nonn

Conjecture 1: a(n) > 0 for all n > 5, and a(n) = 1 only for n = 0, 2, 7, 9, 11, 12, 16, 31, 33, 41.
Conjecture 2: Let n be any integer greater than 9, and let p(x) denote x*(3x+1)/2. For each c = 2, 4, 9, we can write n as p(x) + 2*p(y) + 3*p(z) + c*p(w) with x,y,z,w nonnegative integers.
See also Conjecture 5.2 of the linked 2016 paper.

			a(12) = 1 with 12 = 0*(3*0+1)/2 + 1*(3*1+1)/2 + 1*(3*1+1) + 3*1*(3*1+1)/2.
a(31) = 1 with 31 = 1*(3*1+1)/2 + 3*(3*3+1)/2 + 2*(3*2+1) + 3*0*(3*0+1)/2.
a(33) = 1 with 33 = 2*(3*2+1)/2 + 4*(3*4+1)/2 + 0*(3*0+1) + 3*0*(3*0+1)/2.
a(41) = 1 with 41 = 3*(3*3+1)/2 + 4*(3*4+1)/2 + 0*(3*0+1) + 3*0*(3*0+1)/2.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, A result similar to Lagrange's theorem, J. Number Theory 162(2016), 190-211.

Cf. A005449, A306382, A306383.

PQ[n_]:=PQ[n]=IntegerQ[Sqrt[24n+1]]&&Mod[Sqrt[24n+1],6]==1;
tab={};Do[r=0;Do[If[PQ[n-3x(3x+1)/2-y(3y+1)-z(3z+1)/2],r=r+1],{x,0,(Sqrt[8n+1]-1)/6},{y,0,(Sqrt[12(n-3x(3x+1)/2)+1]-1)/6},{z,0,(Sqrt[12(n-3x(3x+1)/2-y(3y+1))+1]-1)/6}];tab=Append[tab,r],{n,0,100}];Print[tab]

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A306383 Number of ways to write n as x(2x+1) + y(2y+1) + z*(2z+1), where x,y,z are nonnegative integers with x <= y <= z.

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A306439 Number of ways to write n as x(3x+1)/2 + y(3y+1)/2 + z(3z+1) + 3w(3w+1)/2, where x,y,z,w are nonnegative integers with x <= y.

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cp's OEIS Frontend

A306383 Number of ways to write n as x*(2x+1) + y*(2y+1) + z*(2z+1), where x,y,z are nonnegative integers with x <= y <= z.

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A306439 Number of ways to write n as x*(3x+1)/2 + y*(3y+1)/2 + z*(3z+1) + 3w*(3w+1)/2, where x,y,z,w are nonnegative integers with x <= y.

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A306383 Number of ways to write n as x(2x+1) + y(2y+1) + z*(2z+1), where x,y,z are nonnegative integers with x <= y <= z.

A306439 Number of ways to write n as x(3x+1)/2 + y(3y+1)/2 + z(3z+1) + 3w(3w+1)/2, where x,y,z,w are nonnegative integers with x <= y.