

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
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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

ZhiWei Sun, Table of n, a(n) for n = 1..10000
ZhiWei Sun, Refining Lagrange's foursquare theorem, J. Number Theory 175(2017), 167190.
ZhiWei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97120.
ZhiWei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 20172018.


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[mc[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

ZhiWei Sun, Apr 25 2018


STATUS

approved



