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URL: https://oeis.org/A387862

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A387862
Irregular array, read by rows. Let S be the additive monoid generated by the square roots of positive integers, sorted in increasing order. Row n consists of the coefficients of the minimal polynomial of S(n) over the rationals, in ascending order.
1
0, 1, -1, 1, -2, 0, 1, -3, 0, 1, -2, 1, -5, 0, 1, -1, -2, 1, -6, 0, 1, -7, 0, 1, -2, -2, 1, -8, 0, 1, -3, 1, 1, 0, -10, 0, 1, -10, 0, 1, -4, -2, 1, -11, 0, 1, 2, -4, 1, -5, -2, 1, -12, 0, 1, -13, 0, 1, -6, -2, 1, 9, 0, -14, 0, 1, 1, -4, 1, -14, 0, 1, -7, -2, 1, 16, 0, -16, 0, 1, -15, 0, 1, 4, 0
OFFSET
1,5
COMMENTS
Since each S(k) is an algebraic integer, its minimal polynomial is monic with integer coefficients.
The conjugates of Sum_k b_k * sqrt(c_k) are obtained by changing the signs of any nonempty subset of the terms b_k * sqrt(c_k) where c_k > 1.
LINKS
Robert Israel, Table of n, a(n) for n = 1..10572 (rows 1 to 1800, flattened)
EXAMPLE
S(25) = 1 + 2*sqrt(2) has minimal polynomial X^2 - 2*X - 7, so row 25 is [-7, -2, 1].
The first 10 rows are
0, 1;
-1, 1;
-2, 0, 1;
-3, 0, 1;
-2, 1;
-5, 0, 1;
-1, -2, 1;
-6, 0, 1;
-7, 0, 1;
-2, -2, 1.
MAPLE
N:= 5: # for terms before the first appearance of N+1 in S
P[0]:= {0}:
for i from 1 to N do P[i]:= {} od:
for x from 1 to (N+1)^2 do
for i from 0 to N do SP[i]:= {} od:
if not numtheory:-issqrfree(x) then next fi;
for j from 1 to floor((N+1)/sqrt(x)) do
for i from 0 to floor(N +1 - j*sqrt(x)) do
for y in P[i] do
z:= y + j*sqrt(x);
iz:= floor(z);
if iz <= N then SP[iz]:= SP[iz] union {z} fi
od od od;
for i from 0 to N do P[i]:= P[i] union SP[i] od;
od;
for i from 0 to N do
P[i]:= sort(convert(P[i], list), (a, b) -> is(a<b))
od:
PP:= [seq(op(P[i]), i=0..N)]:
g:= proc(t) local p, d, i, X;
p:= evala(Minpoly(t, X));
d:= degree(p, X);
seq(coeff(p, X, i), i=0..d);
end proc:
for i from 1 to nops(PP) do
g(PP[i])
od;
CROSSREFS
Cf. A387686.
Sequence in context: A143987 A309013 A112760 * A096087 A128138 A308999
KEYWORD
sign,tabf
AUTHOR
Robert Israel, Sep 10 2025
STATUS
approved