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A262396
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Product of the sums and differences of the square roots of the first n positive integers, combined in all possible ways.
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2
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OFFSET
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0,4
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COMMENTS
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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, ...
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LINKS
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EXAMPLE
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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)).
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MAPLE
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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))):
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MATHEMATICA
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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];
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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