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 A302982 Number of ways to write n as x^2 + 5*y^2 + 2^z + 3*2^w with x,y,z,w nonnegative integers. 30
 0, 0, 0, 1, 2, 1, 2, 4, 3, 3, 5, 4, 6, 7, 4, 7, 5, 4, 7, 8, 5, 5, 8, 5, 9, 7, 6, 13, 10, 7, 9, 10, 7, 12, 11, 8, 11, 7, 7, 11, 11, 6, 11, 13, 6, 10, 7, 7, 17, 13, 6, 13, 14, 9, 11, 18, 10, 13, 14, 11 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Conjecture: a(n) > 0 for all n > 3. Clearly, a(4*n) > 0 if a(n) > 0. We have verified a(n) > 0 for all n = 4..2*10^8. See also A302983 and A302984 for similar conjectures. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..10000 Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190. Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120. Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018. EXAMPLE a(4) = 1 with 4 = 0^2 + 5*0^2 + 2^0 + 3*2^0. a(5) = 2 with 5 =  1^2 + 5*0^2 + 2^0 + 3*2^0 = 0^2 + 5*0^2 + 2^1 + 3*2^0. a(6) = 1 with 6 = 1^2 + 3*0^2 + 2^1 + 3*2^0. MATHEMATICA SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]; tab={}; Do[r=0; Do[If[SQ[n-3*2^k-2^j-5x^2], r=r+1], {k, 0, Log[2, n/3]}, {j, 0, If[n==3*2^k, -1, Log[2, n-3*2^k]]}, {x, 0, Sqrt[(n-3*2^k-2^j)/5]}]; tab=Append[tab, r], {n, 1, 60}]; Print[tab] CROSSREFS Cf. A000079, A000290, A020669, A271518, A281976, A299924, A299537, A299794, A300219, A300362, A300396, A300441, A301376, A301391, A301471, A301472, A301534, A302920, A302981, A302983, A302984. Sequence in context: A209128 A209131 A165053 * A238577 A131380 A100461 Adjacent sequences:  A302979 A302980 A302981 * A302983 A302984 A302985 KEYWORD nonn AUTHOR Zhi-Wei Sun, Apr 16 2018 STATUS approved

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Last modified February 17 18:46 EST 2019. Contains 320222 sequences. (Running on oeis4.)