A352629 Number of ways to write n as a^2 + 2*b^2 + c^4 + 9*c^4 + 3*c^2*d^2, where a,b,c,d are nonnegative integers.

1, 2, 2, 2, 2, 1, 1, 1, 1, 4, 3, 2, 3, 3, 1, 2, 3, 5, 6, 5, 3, 2, 4, 1, 3, 6, 2, 5, 4, 1, 1, 3, 3, 5, 6, 3, 4, 4, 4, 2, 4, 4, 3, 5, 2, 5, 4, 2, 3, 7, 4, 4, 5, 2, 5, 4, 2, 5, 3, 4, 4, 2, 2, 3, 6, 2, 5, 7, 3, 2, 6, 1, 3, 8, 1, 5, 3, 3, 1, 2, 3, 8, 8, 4, 4, 5, 3, 3, 6, 5, 6, 4, 3, 1, 5, 2, 3, 7, 5, 8, 5

Offset

0,2

Comments

Conjecture: a(n) > 0 for all n = 0,1,2,.... In other words, any nonnegative integer can be written as a^2 + 2*b^2 + c^4 + 9*d^4 + 3*c^2*d^2 with a,b,c,d integers.

It seems that a(n) = 1 only for n = 0, 5, 6, 7, 8, 14, 23, 29, 30, 71, 74, 78, 93, 143, 197, 221, 266, 312, 407, 453, 586, 703, 716, 794.

See also A352627, A352628 and A352632 for similar conjectures.

Links

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

Example

a(30) = 1 with 30 = 3^2 + 2*2^2 + 1^4 + 9*1^4 + 3*1^2*1^2.
a(71) = 1 with 71 = 4^2 + 2*3^2 + 2^4 + 9*1^4 + 3*2^2*1^2.
a(312) = 1 with 312 = 15^2 + 2*5^2 + 2^4 + 9*1^4  + 3*2^2*1^2.
a(703) = 1 with 703 = 26^2 + 2*3^2 + 0^4 + 9*1^4 + 3*0^2*1^2.
a(716) = 1 with 716 = 18^2 + 2*14^2 + 0^4 + 9*0^4 + 3*0^2*0^2.
a(794) = 1 with 794 = 13^2 + 2*0^2 + 5^4 + 9*0^4 + 3*5^2*0^2.

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={}; Do[r=0; Do[If[SQ[n-9d^4-c^4-3c^2*d^2-2b^2], r=r+1], {d, 0, (n/9)^(1/4)}, {c, 0, Sqrt[(Sqrt[4n-27*d^4]-3d^2)/2]}, {b, 0, Sqrt[(n-9d^4-c^4-3c^2*d^2)/2]}]; tab=Append[tab, r], {n, 0, 100}]; Print[tab]

Crossrefs

Cf. A000290, A000583, A352627, A352628, A352632.

Sequence in context: A275301 A282542 A271518 * A106825 A156608 A368752

Adjacent sequences: A352626 A352627 A352628 * A352630 A352631 A352632

Keyword

nonn

Author

Zhi-Wei Sun, Mar 24 2022

Status

approved