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A049816 Triangular array T read by rows: T(n,k) = number of nonzero remainders when Euclidean algorithm acts on n and k, for k=1..n, n>=1.

%I A049816 #19 Nov 29 2023 15:54:10
%S A049816 0,0,0,0,1,0,0,0,1,0,0,1,2,1,0,0,0,0,1,1,0,0,1,1,2,2,1,0,0,0,2,0,3,1,
%T A049816 1,0,0,1,0,1,2,1,2,1,0,0,0,1,1,0,2,2,1,1,0,0,1,2,2,1,2,3,3,2,1,0,0,0,
%U A049816 0,0,2,0,3,1,1,1,1,0,0,1,1,1,3,1,2,4,2,2,2,1,0
%N A049816 Triangular array T read by rows: T(n,k) = number of nonzero remainders when Euclidean algorithm acts on n and k, for k=1..n, n>=1.
%H A049816 Alois P. Heinz, <a href="/A049816/b049816.txt">Rows n = 1..200, flattened</a>
%e A049816 Triangle begins:
%e A049816 0,
%e A049816 0, 0,
%e A049816 0, 1, 0,
%e A049816 0, 0, 1, 0,
%e A049816 0, 1, 2, 1, 0,
%e A049816 0, 0, 0, 1, 1, 0,
%e A049816 0, 1, 1, 2, 2, 1, 0,
%e A049816 0, 0, 2, 0, 3, 1, 1, 0,
%e A049816 0, 1, 0, 1, 2, 1, 2, 1, 0,
%e A049816 0, 0, 1, 1, 0, 2, 2, 1, 1, 0,
%e A049816 0, 1, 2, 2, 1, 2, 3, 3, 2, 1, 0,
%e A049816 0, 0, 0, 0, 2, 0, 3, 1, 1, 1, 1, 0,
%e A049816 0, 1, 1, 1, 3, 1, 2, 4, 2, 2, 2, 1, 0,
%e A049816 ...
%p A049816 T:= proc(x, y) option remember;
%p A049816       `if`(y=0, -1, 1+T(y, irem(x, y)))
%p A049816     end:
%p A049816 seq(seq(T(n, k), k=1..n), n=1..15);  # _Alois P. Heinz_, Nov 29 2023
%t A049816 R[n_, k_] := R[n, k] = With[{r = Mod[n, k]}, If[r == 0, 1, R[k, r] + 1]];
%t A049816 T[n_, k_] := R[n, k] - 1;
%t A049816 Table[T[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Apr 12 2019, after _Robert Israel_ in A107435 *)
%Y A049816 Row sums give A049817.
%K A049816 nonn,tabl
%O A049816 1,13
%A A049816 _Clark Kimberling_