A049828 Triangular array T given by rows: T(n,k)=sum of remainders when Euclidean algorithm acts on n,k; for k=1,2,...,n; n >= 1.

0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 3, 1, 0, 0, 0, 0, 2, 1, 0, 0, 1, 1, 4, 3, 1, 0, 0, 0, 3, 0, 6, 2, 1, 0, 0, 1, 0, 1, 5, 3, 3, 1, 0, 0, 0, 1, 2, 0, 6, 4, 2, 1, 0, 0, 1, 3, 4, 1, 6, 8, 6, 3, 1, 0, 0, 0, 0, 0, 3, 0, 8, 4, 3, 2, 1, 0, 0, 1, 1, 1, 6, 1, 7, 11, 5, 4, 3, 1, 0

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• tabl

Comments

For a fixed n, {(k,T(n,k)), k=1..n} is conjectured to be fractal in nature (see link). - Tiberiu Szocs-Mihai, Aug 17 2015

Examples

			Rows:
0;
0, 0;
0, 1, 0;
0, 0, 1, 0;
0, 1, 3, 1, 0;
0, 0, 0, 2, 1, 0;
0, 1, 1, 4, 3, 1, 0;
...
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10011 (rows 1 to 141, flattened)
• Tiberiu Szocs-Mihai, Euclidean fractal (conjectured), Math Ticks Blog, January 2011.
• Tiberiu Szocs-Mihai, Euclidean fractal candidate description, Math Ticks Blog, January 2011.
• Eric Weisstein's World of Mathematics, Euclidean Algorithm.

Crossrefs

Row sums are in A049829.

Programs

Maple

T:=  proc(n,k) option remember;
if n*k = 0 then 0 else (n mod k) + procname(k,n mod k) fi
end proc:
seq(seq(T(n,k),k=1..n), n=1..20); # Robert Israel, Aug 31 2015

Mathematica

T[n_, k_] := T[n, k] = If[n*k == 0, 0, Mod[n, k] + T[k, Mod[n, k]]];
Table[T[n, k], {n, 1, 20}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 27 2019, after Robert Israel *)

PARI

tabl(nn) = {for (n=1, nn, for (k=1, n, a = n; b = k; r = 1; s = 0; while (r, q = a\b; r = a - b*q; s +=r; a = b; b = r); print1(s, ", ");); print(););} \\ Michel Marcus, Aug 17 2015