A262396 Product of the sums and differences of the square roots of the first n positive integers, combined in all possible ways.

1, -1, 1, 64, 4096, 23323703841, 63703464216016403230349121, 316699666163357097153212433469030615484754548657341071360000

Offset: 0

Author

Mark Bradley, Sep 21 2015

Keywords

• sign

Comments

The series increases rapidly and the next number in the sequence has 135 decimal digits. Each element for n>1 is necessarily both an integer and a perfect square, the square roots being 1, 8, 64, 152721, 7981444995489, 562760753929551396141111705600, ...

Examples

			a(0) = 1 = (empty product).
a(1) = -1 = (sqrt(1)) * (-sqrt(1)).
a(2) = 1 = (1+sqrt(2)) * (1-sqrt(2)) * (-1+sqrt(2)) * (-1-sqrt(2)).
a(3) = 64 = (1+sqrt(2)+sqrt(3)) * (1+sqrt(2)-sqrt(3)) * (1-sqrt(2)+sqrt(3)) * (1-sqrt(2)-sqrt(3)) * (-1+sqrt(2)+sqrt(3)) * (-1+sqrt(2)-sqrt(3)) * (-1-sqrt(2)+sqrt(3)) * (-1-sqrt(2)-sqrt(3)).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10

Crossrefs

Cf. A354913.

Programs

Maple

s:= proc(n) option remember; `if`(n<2, [1, -1][1..2*n],
       map(x-> [x+sqrt(n), x-sqrt(n)][], s(n-1)))
    end:
a:= n-> expand(mul(t, t=s(n))):
seq(a(n), n=0..7);  # Alois P. Heinz, Sep 21 2015

Mathematica

s[n_] := s[n] = If[n < 2, {1, -1}[[1 ;; 2n]], {# + Sqrt[n], # - Sqrt[n]}& /@ s[n - 1]];
a[n_] := If[n == 0, 1, Times @@ Flatten[s[n], n - 1] // Expand];
a /@ Range[0, 7] (* Jean-François Alcover, Nov 24 2020, after Alois P. Heinz *)