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A352628 Number of ways to write n as a^2 + 2*b^2 + c^4 + 2*d^4 + 3*c^2*d^2, where a,b,c,d are nonnegative integers. 5
1, 2, 3, 3, 3, 2, 3, 2, 3, 4, 4, 3, 3, 2, 2, 2, 2, 4, 6, 5, 4, 1, 3, 2, 5, 5, 2, 4, 4, 2, 2, 2, 4, 8, 8, 5, 5, 2, 7, 5, 4, 5, 4, 5, 4, 3, 3, 3, 6, 8, 7, 6, 6, 3, 8, 4, 5, 9, 2, 6, 4, 2, 2, 6, 5, 5, 7, 6, 7, 3, 6, 1, 6, 8, 5, 4, 3, 3, 6, 3, 3, 10, 9, 10, 6, 2, 4, 7, 6, 9, 4, 3, 3, 2, 3, 2, 7, 8, 9, 12, 8 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Conjecture: a(n) > 0 for all n = 0,1,2,.... In other words, each nonnegative integer can be written as a^2 + 2*b^2 + (c^2+d^2)*(c^2+2*d^2) with a,b,c,d integers.
It seems that a(n) = 1 only for n = 0, 21, 71, 157, 175, 190, 316, 476, 526.
See also A352627, A352629 and A352632 for similar conjectures.
LINKS
EXAMPLE
a(21) = 1 with 21 = 1^2 + 2*3^2 + 0^4 +2*1^4 + 3*0^2*1^2.
a(71) = 1 with 71 = 3^2 + 2*4^2 + 2^4 + 2*1^4 + 3*2^2*1^2.
a(157) = 1 with 157 = 2^2 + 2*6^2 + 3^4 + 2*0^4 + 3*3^2*0^2.
a(175) = 1 with 175 = 13^2 + 2*0^2 + 1^4 + 2*1^4 + 3*1^2*1^2.
a(190) = 1 with 190 = 0^2 + 2*0^2 + 1^4 + 2*3^4 + 3*1^2*3^2.
a(316) = 1 with 316 = 10^2 + 2*10^2 + 2^4 + 2*0^4 + 3*2^2*0^2.
a(476) = 1 with 476 = 5^2 + 2*15^2 + 1^4 + 2*0^4 + 3*1^2*0^2.
a(526) = 1 with 526 = 18^2 + 2*10^2 + 0^4 + 2*1^4 + 3*0^2*1^2.
MATHEMATICA
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={}; Do[r=0; Do[If[SQ[n-2d^4-c^4-3c^2*d^2-2b^2], r=r+1], {d, 0, (n/2)^(1/4)}, {c, 0, Sqrt[(Sqrt[4n+d^4]-3d^2)/2]}, {b, 0, Sqrt[(n-2d^4-c^4-3c^2*d^2)/2]}]; tab=Append[tab, r], {n, 0, 100}]; Print[tab]
CROSSREFS
Sequence in context: A366544 A096520 A236552 * A080748 A305374 A084966
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Mar 24 2022
STATUS
approved

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Last modified April 21 09:27 EDT 2024. Contains 371851 sequences. (Running on oeis4.)