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

1,2

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