A352627 Number of ways to write n as a^2 + 2*b^2 + c^4 + 4*d^4 + c^2*d^2, where a,b,c,d are nonnegative integers.

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

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^4 + 4*d^4 + c^2*d^2 with a,b,c,d integers.

It seems that a(n) = 1 only for n = 0, 11, 14, 21, 29, 46, 53, 62, 149, 174, 221, 239, 254, 1039, 1709, 2239.

See also A352628, A352629 and A352632 for similar conjectures.

Links

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

Example

a(11) = 1 with 11 = 3^2 + 2*1^2 + 0^4 + 4*0^4 + 0^2*0^2.
a(14) = 1 with 14 = 0^2 + 2*2^2 + 1^4 + 4*1^4 + 1^2*1^2.
a(221) = 1 with 221 = 12^2 + 2*2^2 + 1^4 + 4*2^4 + 1^2*2^2.
a(239) = 1 with 239 = 15^2 + 2*2^2 + 1^4 + 4*1^4 + 1^2*1^2.
a(254) = 1 with 254 = 1^2 + 2*6^2 + 3^4 + 4*2^4 + 3^2*2^2.
a(1039) = 1 with 1039 = 31^2 + 2*6^2 + 1^4 + 4*1^4 + 1^2*1^2.
a(1709) = 1 with 1709 = 9^2 + 2*26^2 + 4^4 + 4*1^4 + 4^2*1^2.
a(2239) = 1 with 2239 = 41^2 + 2*6^2 + 3^4 + 4*3^4 + 3^2*3^2.

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={}; Do[r=0; Do[If[SQ[n-4d^4-c^4-c^2*d^2-2b^2], r=r+1], {d, 0, (n/4)^(1/4)}, {c, 0, Sqrt[(Sqrt[4n-15*d^4]-d^2)/2]}, {b, 0, Sqrt[(n-4d^4-c^4-c^2*d^2)/2]}]; tab=Append[tab, r], {n, 0, 100}]; Print[tab]

Crossrefs

Cf. A000290, A000583, A352628, A352629, A352632.

Sequence in context: A342739 A137734 A366066 * A182210 A078705 A346018

Adjacent sequences: A352624 A352625 A352626 * A352628 A352629 A352630

Keyword

nonn

Author

Zhi-Wei Sun, Mar 24 2022

Status

approved