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 A273404 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with x + 24*y a square, where x,y,z,w are nonnegative integers with z <= w. 11
 1, 2, 3, 2, 2, 3, 3, 2, 1, 3, 4, 2, 1, 2, 2, 2, 2, 3, 5, 2, 3, 3, 2, 1, 1, 4, 5, 4, 2, 2, 4, 3, 3, 3, 6, 2, 6, 5, 3, 3, 3, 7, 6, 2, 2, 5, 4, 1, 2, 3, 7, 6, 8, 4, 5, 5, 2, 4, 5, 2, 3, 5, 3, 4, 2, 5, 9, 4, 5, 4, 5, 1, 3, 5, 4, 5, 5, 4, 2, 3, 3 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Conjecture: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 16^k*m (k = 0,1,2,... and m = 8, 12, 23, 24, 47, 71, 168, 344, 632). For more conjectural refinements of Lagrange's four-square theorem, one may consult arXiv:1604.06723. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 0..10000 Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.GM], 2016. EXAMPLE a(8) = 1 since 8 = 0^2 + 0^2 + 2^2 + 2^2 with 0 + 24*0 = 0^2. a(12) = 1 since 12 = 1^2 + 1^2 + 1^2 + 3^2 with 1 + 24*1 = 5^2. a(23) = 1 since 23 = 1^2 + 2^2 + 3^2 + 3^2 with 1 + 24*2 = 7^2. a(24) = 1 since 24 = 4^2 + 0^2 + 2^2 + 2^2 with 4 + 24*0 = 2^2. a(47) = 1 since 47 = 1^2 + 1^2 + 3^2 + 6^2 with 1 + 24*1 = 5^2. a(71) = 1 since 71 = 1^2 + 5^2 + 3^2 + 6^2 with 1 + 24*5 = 11^2. a(168) = 1 since 168 = 4^2 + 4^2 + 6^2 + 10^2 with 4 + 24*4 = 10^2. a(344) = 1 since 344 = 4^2 + 0^2 + 2^2 + 18^2 with 4 + 24*0 = 2^2. a(632) = 1 since 632 = 0^2 + 6^2 + 14^2 + 20^2 with 0 + 24*6 = 12^2. MATHEMATICA SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]] Do[r=0; Do[If[SQ[n-x^2-y^2-z^2]&&SQ[x+24y], r=r+1], {x, 0, Sqrt[n]}, {y, 0, Sqrt[n-x^2]}, {z, 0, Sqrt[(n-x^2-y^2)/2]}]; Print[n, " ", r]; Label[aa]; Continue, {n, 0, 80}] CROSSREFS Cf. A000118, A000290, A260625, A261876, A262357, A267121, A268197, A268507, A269400, A270073, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272332, A272351, A272620, A272888, A272977, A273021, A273107, A273108, A273110, A273134, A273278, A273294, A273302. Sequence in context: A126014 A317420 A256795 * A281976 A300708 A240755 Adjacent sequences:  A273401 A273402 A273403 * A273405 A273406 A273407 KEYWORD nonn AUTHOR Zhi-Wei Sun, May 21 2016 STATUS approved

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Last modified July 12 02:02 EDT 2020. Contains 335658 sequences. (Running on oeis4.)