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A232209
Height of algebraic number 1 + sqrt(2) + ... + sqrt(n).
1
1, 2, 16, 48, 10140, 6552, 721125376, 3620732928, 278799279816, 29925033224, 229926056690973293936640, 892398340719534485274624, 603207249820766251389767637583758341569376980491272, 240171846906336440253785749946778562802349467993472
OFFSET
1,2
COMMENTS
Let p(z) be the monic minimal polynomial of sum(j=1..n, sqrt(j)) over the integers.
a(n) is the maximum of the absolute values of the coefficients of p(z).
The degree of p(z) is at most A048656(n). Is it always equal to A048656(n)?
LINKS
Springer, Encyclopedia of Mathematics, Algebraic number
EXAMPLE
For n = 1 the minimal polynomial of 1 is z - 1 so a(1) = 1.
For n = 2 the minimal polynomial of 1 + sqrt(2) is z^2 - 2*z - 1 so a(2) = 2.
MAPLE
for n from 1 to 15 do
a:= convert(add(sqrt(i), i=1..n), RootOf);
P:= evala(Norm(a-z));
A[n]:= max(map(abs, [coeffs(P, z)]));
od:
seq(A[n], n=1..15); # Robert Israel, Sep 10 2014
MATHEMATICA
a[n_] := CoefficientList[ MinimalPolynomial[ Sqrt[Range[n]] // Total, x], x] // Abs // Max; Array[a, 12] (* Jean-François Alcover, Apr 29 2019 *)
PROG
(GAP) a:=function(n)
return MinimalPolynomial(Rationals, Sum([1..n], x->Sqrt(x)));
end; # Charles R Greathouse IV, Sep 12 2014
CROSSREFS
Cf. A048656.
Sequence in context: A159010 A223219 A063721 * A012180 A058376 A295906
KEYWORD
nonn
AUTHOR
Robert Israel, Sep 10 2014
STATUS
approved