login
A345415
Table read by upward antidiagonals: Given m, n >= 1, write gcd(m,n) as d = u*m+v*n where u, v are minimal; T(m,n) = u.
18
0, 0, 1, 0, 0, 1, 0, 1, -1, 1, 0, 0, 0, 1, 1, 0, 1, 1, -1, -2, 1, 0, 0, -1, 0, 2, 1, 1, 0, 1, 0, 1, -1, 1, -3, 1, 0, 0, 1, 1, 0, -1, -2, 1, 1, 0, 1, -1, -1, 1, -1, 2, 3, -4, 1, 0, 0, 0, 0, -2, 0, 3, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, -1, -3, -2, -3, -5, 1, 0, 0, -1, 1, -1, 1, 0, -1, 2, -2, 4, 1, 1
OFFSET
1,20
COMMENTS
The gcd is given in A003989, and v is given in A345416. Minimal means minimize u^2+v^2. We follow Maple, PARI, etc., in setting u=0 and v=1 when m=n. If we ignore the diagonal, the v table is the transpose of the u table.
EXAMPLE
The gcd table (A003989) begins:
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
[1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2]
[1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1]
[1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4]
[1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1]
[1, 2, 3, 2, 1, 6, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2]
[1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1]
[1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 8]
[1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1]
[1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 2, 1, 2, 5, 2]
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1]
[1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 12, 1, 2, 3, 4]
...
The u table (this entry) begins:
[0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
[0, 0, -1, 1, -2, 1, -3, 1, -4, 1, -5, 1, -6, 1, -7, 1]
[0, 1, 0, -1, 2, 1, -2, 3, 1, -3, 4, 1, -4, 5, 1, -5]
[0, 0, 1, 0, -1, -1, 2, 1, -2, -2, 3, 1, -3, -3, 4, 1]
[0, 1, -1, 1, 0, -1, 3, -3, 2, 1, -2, 5, -5, 3, 1, -3]
[0, 0, 0, 1, 1, 0, -1, -1, -1, 2, 2, 1, -2, -2, -2, 3]
[0, 1, 1, -1, -2, 1, 0, -1, 4, 3, -3, -5, 2, 1, -2, 7]
[0, 0, -1, 0, 2, 1, 1, 0, -1, -1, -4, -1, 5, 2, 2, 1]
[0, 1, 0, 1, -1, 1, -3, 1, 0, -1, 5, -1, 3, -3, 2, -7]
[0, 0, 1, 1, 0, -1, -2, 1, 1, 0, -1, -1, 4, 3, -1, -3]
[0, 1, -1, -1, 1, -1, 2, 3, -4, 1, 0, -1, 6, -5, -4, 3]
[0, 0, 0, 0, -2, 0, 3, 1, 1, 1, 1, 0, -1, -1, -1, -1]
...
The v table (A345416) begins:
[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
[1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0]
[1, -1, 1, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1]
[1, 1, -1, 1, 1, 1, -1, 0, 1, 1, -1, 0, 1, 1, -1, 0]
[1, -2, 2, -1, 1, 1, -2, 2, -1, 0, 1, -2, 2, -1, 0, 1]
[1, 1, 1, -1, -1, 1, 1, 1, 1, -1, -1, 0, 1, 1, 1, -1]
[1, -3, -2, 2, 3, -1, 1, 1, -3, -2, 2, 3, -1, 0, 1, -3]
[1, 1, 3, 1, -3, -1, -1, 1, 1, 1, 3, 1, -3, -1, -1, 0]
[1, -4, 1, -2, 2, -1, 4, -1, 1, 1, -4, 1, -2, 2, -1, 4]
[1, 1, -3, -2, 1, 2, 3, -1, -1, 1, 1, 1, -3, -2, 1, 2]
[1, -5, 4, 3, -2, 2, -3, -4, 5, -1, 1, 1, -5, 4, 3, -2]
[1, 1, 1, 1, 5, 1, -5, -1, -1, -1, -1, 1, 1, 1, 1, 1]
...
MAPLE
mygcd:=proc(a, b) local d, s, t; d := igcdex(a, b, `s`, `t`); [a, b, d, s, t]; end;
gcd_rowu:=(m, M)->[seq(mygcd(m, n)[4], n=1..M)];
for m from 1 to 12 do lprint(gcd_rowu(m, 16)); od;
MATHEMATICA
T[m_, n_] := Module[{u, v}, MinimalBy[{u, v} /. Solve[u^2 + v^2 <= 26 && u*m + v*n == GCD[m, n], {u, v}, Integers], #.#&][[1, 1]]];
Table[T[m - n + 1, n], {m, 1, 13}, {n, 1, m}] // Flatten (* Jean-François Alcover, Mar 27 2023 *)
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
N. J. A. Sloane, Jun 19 2021
STATUS
approved