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A345416 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) = v.

%I A345416 #19 Mar 27 2023 12:44:59
%S A345416 1,1,0,1,1,0,1,-1,1,0,1,1,1,0,0,1,-2,-1,1,1,0,1,1,2,1,-1,0,0,1,-3,1,
%T A345416 -1,1,0,1,0,1,1,-2,-1,1,1,1,0,0,1,-4,3,2,-1,1,-1,-1,1,0,1,1,1,1,3,1,
%U A345416 -2,0,0,0,0,1,-5,-3,-2,-3,-1,1,2,1,1,1,0,1,1,4,-2,2,-1,1,1,-1,1,-1,0,0
%N A345416 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) = v.
%C A345416 The gcd is given in A003989, and u is given in A345415. 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.
%e A345416 The gcd table (A003989) begins:
%e A345416 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
%e A345416 [1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2]
%e A345416 [1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1]
%e A345416 [1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4]
%e A345416 [1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1]
%e A345416 [1, 2, 3, 2, 1, 6, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2]
%e A345416 [1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1]
%e A345416 [1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 8]
%e A345416 [1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1]
%e A345416 [1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 2, 1, 2, 5, 2]
%e A345416 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1]
%e A345416 [1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 12, 1, 2, 3, 4]
%e A345416 ...
%e A345416 The u table (A345415) begins:
%e A345416 [0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
%e A345416 [0, 0, -1, 1, -2, 1, -3, 1, -4, 1, -5, 1, -6, 1, -7, 1]
%e A345416 [0, 1, 0, -1, 2, 1, -2, 3, 1, -3, 4, 1, -4, 5, 1, -5]
%e A345416 [0, 0, 1, 0, -1, -1, 2, 1, -2, -2, 3, 1, -3, -3, 4, 1]
%e A345416 [0, 1, -1, 1, 0, -1, 3, -3, 2, 1, -2, 5, -5, 3, 1, -3]
%e A345416 [0, 0, 0, 1, 1, 0, -1, -1, -1, 2, 2, 1, -2, -2, -2, 3]
%e A345416 [0, 1, 1, -1, -2, 1, 0, -1, 4, 3, -3, -5, 2, 1, -2, 7]
%e A345416 [0, 0, -1, 0, 2, 1, 1, 0, -1, -1, -4, -1, 5, 2, 2, 1]
%e A345416 [0, 1, 0, 1, -1, 1, -3, 1, 0, -1, 5, -1, 3, -3, 2, -7]
%e A345416 [0, 0, 1, 1, 0, -1, -2, 1, 1, 0, -1, -1, 4, 3, -1, -3]
%e A345416 [0, 1, -1, -1, 1, -1, 2, 3, -4, 1, 0, -1, 6, -5, -4, 3]
%e A345416 [0, 0, 0, 0, -2, 0, 3, 1, 1, 1, 1, 0, -1, -1, -1, -1]
%e A345416 ...
%e A345416 The v table (this entry) begins:
%e A345416 [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
%e A345416 [1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0]
%e A345416 [1, -1, 1, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1]
%e A345416 [1, 1, -1, 1, 1, 1, -1, 0, 1, 1, -1, 0, 1, 1, -1, 0]
%e A345416 [1, -2, 2, -1, 1, 1, -2, 2, -1, 0, 1, -2, 2, -1, 0, 1]
%e A345416 [1, 1, 1, -1, -1, 1, 1, 1, 1, -1, -1, 0, 1, 1, 1, -1]
%e A345416 [1, -3, -2, 2, 3, -1, 1, 1, -3, -2, 2, 3, -1, 0, 1, -3]
%e A345416 [1, 1, 3, 1, -3, -1, -1, 1, 1, 1, 3, 1, -3, -1, -1, 0]
%e A345416 [1, -4, 1, -2, 2, -1, 4, -1, 1, 1, -4, 1, -2, 2, -1, 4]
%e A345416 [1, 1, -3, -2, 1, 2, 3, -1, -1, 1, 1, 1, -3, -2, 1, 2]
%e A345416 [1, -5, 4, 3, -2, 2, -3, -4, 5, -1, 1, 1, -5, 4, 3, -2]
%e A345416 [1, 1, 1, 1, 5, 1, -5, -1, -1, -1, -1, 1, 1, 1, 1, 1]
%e A345416 ...
%p A345416 mygcd:=proc(a,b) local d,s,t; d := igcdex(a,b,`s`,`t`); [a,b,d,s,t]; end;
%p A345416 gcd_rowv:=(m,M)->[seq(mygcd(m,n)[5],n=1..M)];
%p A345416 for m from 1 to 12 do lprint(gcd_rowv(m,16)); od;
%t A345416 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, 2]]];
%t A345416 Table[T[m - n + 1, n], {m, 1, 13}, {n, 1, m}] // Flatten (* _Jean-François Alcover_, Mar 27 2023 *)
%Y A345416 Cf. A003989, A050873, A345415, A345417, A345418.
%K A345416 sign,tabl
%O A345416 1,17
%A A345416 _N. J. A. Sloane_, Jun 19 2021