|
%I #20 Jul 20 2019 21:04:05
%S 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,
%T 51,55,56,65,66,69,70,71,74,79,82,83,84,86,87,95,101,102,106,119,120,
%U 122,123,127,134
%N Numbers of the form k^2 + binomial(2*m,m) with k and m nonnegative integers.
%C 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.
%C This has been verified for all n = 2..10^10.
%H Zhi-Wei Sun, <a href="/A303541/b303541.txt">Table of n, a(n) for n = 1..10000</a>
%H Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2016.11.008">Refining Lagrange's four-square theorem</a>, J. Number Theory 175(2017), 167-190.
%H Zhi-Wei Sun, <a href="http://maths.nju.edu.cn/~zwsun/179b.pdf">New conjectures on representations of integers (I)</a>, Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1701.05868">Restricted sums of four squares</a>, arXiv:1701.05868 [math.NT], 2017-2018.
%e a(1) = 1 with 0^2 + binomial(2*0,0) = 1.
%e a(7) = 10 with 2^2 + binomial(2*2,2) = 10.
%e a(8) = 11 with 3^2 + binomial(2*1,1) = 11.
%t c[n_]:=c[n]=Binomial[2n,n];
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t 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]
%Y Cf. A000290, A000984, A001481, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432, A303434, A303539, A303540, A303543.
%K nonn
%O 1,2
%A _Zhi-Wei Sun_, Apr 25 2018
|
