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%I #34 Nov 25 2023 08:35:42
%S 1,2,1,2,3,3,1,3,3,4,2,4,1,4,2,4,5,4,4,5,1,5,4,4,5,6,3,2,3,6,1,6,3,2,
%T 3,6,7,5,5,5,5,7,1,7,5,5,5,5,7,8,4,5,2,5,4,8,1,8,4,5,2,5,4,8,9,6,3,6,
%U 6,3,6,9,1,9,6,3,6,6,3,6,9,10,5,6,4,2,4,6,5,10,1,10,5,6,4,2,4,6,5
%N Irregular triangle read by rows: successive bottom and right-hand borders of the infinite square array in A072030 (which gives number of subtraction steps needed to compute GCD).
%C Officially the borders are read starting at the bottom left, reading horizontally until the main diagonal is reached, and then reading vertically upwards until the top row is reached.
%C However, in this case both borders are symmetric about their midpoints, and the bottom border is the same as the right-hand border, so the direction in which the borders are read is less critical.
%H R. J. Mathar, <a href="/A267177/b267177.txt">Table of n, a(n) for n = 1..10000</a>
%e The array in A072030 begins:
%e 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
%e 2, 1, 3, 2, 4, 3, 5, 4, 6, 5, ...
%e 3, 3, 1, 4, 4, 2, 5, 5, 3, 6, ...
%e 4, 2, 4, 1, 5, 3, 5, 2, 6, 4, ...
%e 5, 4, 4, 5, 1, 6, 5, 5, 6, 2, ...
%e 6, 3, 2, 3, 6, 1, 7, 4, 3, 4, ...
%e 7, 5, 5, 5, 5, 7, 1, 8, 6, 6, ...
%e 8, 4, 5, 2, 5, 4, 8, 1, 9, 5, ...
%e 9, 6, 3, 6, 6, 3, 6, 9, 1, 10, ...
%e 10, 5, 6, 4, 2, 4, 6, 5, 10, 1, ...
%e ...
%e The successive bottom and right-hand borders are:
%e 1,
%e 2, 1, 2,
%e 3, 3, 1, 3, 3,
%e 4, 2, 4, 1, 4, 2, 4,
%e 5, 4, 4, 5, 1, 5, 4, 4, 5,
%e 6, 3, 2, 3, 6, 1, 6, 3, 2, 3, 6,
%e 7, 5, 5, 5, 5, 7, 1, 7, 5, 5, 5, 5, 7,
%e ...
%p A267177 := proc(n,k)
%p if k <= n then
%p A072030(n,k) ;
%p else
%p A072030(2*n-k,n) ;
%p end if;
%p end proc:
%p seq(seq(A267177(n,k),k=1..2*n-1),n=1..10) ; # _R. J. Mathar_, May 07 2016
%t A072030[n_, k_] := A072030[n, k] = Which[n < 1 || k < 1, 0, n == k, 1, n < k, A072030[k, n], True, 1+A072030[k, n-k]];
%t A267177[n_, k_] := If[k <= n, A072030[n, k], A072030[2n-k, n]];
%t Table[A267177[n, k], {n, 1, 10}, {k, 1, 2n-1}] // Flatten (* _Jean-François Alcover_, Apr 23 2023, after _R. J. Mathar_ *)
%o (PARI) \\ Based on _Michel Marcus_'s program for A049834.
%o tabl(nn) = {for (n=1, nn,
%o for (k=1, n, a = n; b = k; r = 1; s = 0; while (r, q = a\b; r = a - b*q; s += q; a = b; b = r); print1(s, ", "); );
%o for (k=1, n-1, a = n; b = n-k; r = 1; s = 0; while (r, q = a\b; r = a - b*q; s += q; a = b; b = r); print1(s, ", "); );
%o print(); ); }
%o tabl(12)
%Y Cf. A072030, A049834, A267178 (parity).
%K nonn,tabf,easy
%O 1,2
%A _N. J. A. Sloane_, Jan 14 2016
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