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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.

%I A049828 #22 Feb 16 2025 08:32:40
%S A049828 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,
%T A049828 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,
%U A049828 0,0,3,0,8,4,3,2,1,0,0,1,1,1,6,1,7,11,5,4,3,1,0
%N 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.
%C A049828 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
%H A049828 Robert Israel, <a href="/A049828/b049828.txt">Table of n, a(n) for n = 1..10011</a> (rows 1 to 141, flattened)
%H A049828 Tiberiu Szocs-Mihai, <a href="http://mathticks.blogspot.ro/2011/01/discrete-connections-part-ii.html">Euclidean fractal (conjectured)</a>, Math Ticks Blog, January 2011.
%H A049828 Tiberiu Szocs-Mihai, <a href="http://mathticks.blogspot.ro/2011/01/discrete-connections-part-v.html">Euclidean fractal candidate description</a>, Math Ticks Blog, January 2011.
%H A049828 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EuclideanAlgorithm.html">Euclidean Algorithm.</a>
%e A049828 Rows:
%e A049828 0;
%e A049828 0, 0;
%e A049828 0, 1, 0;
%e A049828 0, 0, 1, 0;
%e A049828 0, 1, 3, 1, 0;
%e A049828 0, 0, 0, 2, 1, 0;
%e A049828 0, 1, 1, 4, 3, 1, 0;
%e A049828 ...
%p A049828 T:=  proc(n,k) option remember;
%p A049828 if n*k = 0 then 0 else (n mod k) + procname(k,n mod k) fi
%p A049828 end proc:
%p A049828 seq(seq(T(n,k),k=1..n), n=1..20); # _Robert Israel_, Aug 31 2015
%t A049828 T[n_, k_] := T[n, k] = If[n*k == 0, 0, Mod[n, k] + T[k, Mod[n, k]]];
%t A049828 Table[T[n, k], {n, 1, 20}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Feb 27 2019, after _Robert Israel_ *)
%o A049828 (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
%Y A049828 Row sums are in A049829.
%K A049828 nonn,tabl
%O A049828 1,13
%A A049828 _Clark Kimberling_