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

%I #15 Jun 22 2024 22:25:03

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

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

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

%F a(n) = sqrt(A000125(A047694(n))), where sqrt = A000194 or A000196. - _M. F. Hasler_, Jun 22 2024

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

%t 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 *)

%o (PARI) A047695=[y|n<-[-1..777], issquare(binomial(n+1, 3)+n+1,&y)] \\ _M. F. Hasler_, Jun 22 2024

%Y Cf. A047694 (the corresponding x values).

%K nonn,fini,full,nice

%O 0,3

%A _N. J. A. Sloane_