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.

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

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

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000

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

Cf. A000290, A000583, A352627, A352629, A352632.

Sequence in context: A366544 A096520 A236552 * A080748 A305374 A084966

Adjacent sequences: A352625 A352626 A352627 * A352629 A352630 A352631

Keyword

nonn

Author

Zhi-Wei Sun, Mar 24 2022

Status

approved