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 A303541 Numbers of the form k^2 + binomial(2*m,m) with k and m nonnegative integers. 22
 1, 2, 3, 5, 6, 7, 10, 11, 15, 17, 18, 20, 21, 22, 24, 26, 27, 29, 31, 36, 37, 38, 42, 45, 50, 51, 55, 56, 65, 66, 69, 70, 71, 74, 79, 82, 83, 84, 86, 87, 95, 101, 102, 106, 119, 120, 122, 123, 127, 134 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The conjecture in A303540 has the following equivalent version: Each integer n > 1 can be written as the sum of two terms of the current sequence. This has been verified for all n = 2..10^10. 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(1) = 1 with 0^2 + binomial(2*0,0) = 1. a(7) = 10 with 2^2 + binomial(2*2,2) = 10. a(8) = 11 with 3^2 + binomial(2*1,1) = 11. MATHEMATICA c[n_]:=c[n]=Binomial[2n, n]; SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]; tab={}; n=0; Do[k=0; Label[bb]; If[c[k]>m, Goto[aa]]; If[SQ[m-c[k]], n=n+1; tab=Append[tab, m]; Goto[aa], k=k+1; Goto[bb]]; Label[aa], {m, 1, 134}]; Print[tab] CROSSREFS Cf. A000290, A000984, A001481, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432, A303434, A303539, A303540, A303543. Sequence in context: A031941 A043089 A088952 * A293523 A032851 A102830 Adjacent sequences:  A303538 A303539 A303540 * A303542 A303543 A303544 KEYWORD nonn AUTHOR Zhi-Wei Sun, Apr 25 2018 STATUS approved

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Last modified August 4 09:03 EDT 2020. Contains 336201 sequences. (Running on oeis4.)