A047695 Nonnegative y such that y^2 = C(x,0) + C(x,1) + C(x,2) + C(x,3) is soluble in integers.

0, 1, 2, 8, 24, 260, 8672

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• fini
• full
• nice

Examples

			For y = 8, we have 8^2 = 64 = 1 + 7 + 21 + 35 = Sum_{k=0..3} C(x, k) for x = 7.
		

References

• R. K. Guy, Unsolved Problems in Number Theory, Section D3.

Crossrefs

Cf. A047694 (the corresponding x values).

Programs

Mathematica

r[x_] := Reduce[y >= 0 && 6*y^2 == (x + 1)*(x^2 - x + 6), {y}, Integers]; Reap[ Do[ If[r[x] =!= False, Sow[y /. ToRules[r[x]]]], {x, -10, 1000}]][[2, 1]] (* Jean-François Alcover, Jul 12 2012 *)

PARI

A047695=[y|n<-[-1..777], issquare(binomial(n+1, 3)+n+1,&y)] \\ M. F. Hasler, Jun 22 2024

Formula

a(n) = sqrt(A000125(A047694(n))), where sqrt = A000194 or A000196. - M. F. Hasler, Jun 22 2024