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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. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,43
COMMENTS
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.
LINKS
Zhi-Wei Sun, Universal sums of three quadratic polynomials, Sci. China Math., in press.
EXAMPLE
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).
MATHEMATICA
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]
CROSSREFS
Sequence in context: A370815 A056137 A332602 * A172099 A170957 A178725
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Feb 11 2019
STATUS
approved

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Last modified July 13 12:36 EDT 2024. Contains 374284 sequences. (Running on oeis4.)