A303541 Numbers of the form k^2 + binomial(2*m,m) with k and m nonnegative integers.

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

Offset: 1

Author

Zhi-Wei Sun, Apr 25 2018

Keywords

• nonn

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.

Examples

			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.
		

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.

Crossrefs

Cf. A000290, A000984, A001481, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432, A303434, A303539, A303540, A303543.

Programs

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]