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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. 5
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 (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, 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
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
Sequence in context: A275301 A282542 A271518 * A106825 A156608 A368752
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Mar 24 2022
STATUS
approved

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Last modified April 18 15:16 EDT 2024. Contains 371780 sequences. (Running on oeis4.)