%I #16 Apr 23 2023 12:18:23
%S 1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,
%T 1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0,1,0,0,1,0,1,0,
%U 0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0,0,0,1,0,1,0,1,0,0,0,0,0,1,0,1,1,0,0,1,1,0
%N Irregular triangle read by rows: successive bottom and right-hand borders of the infinite square array in A072030 (number of subtraction steps needed to compute GCD) read mod 2.
%H R. J. Mathar, <a href="/A267179/b267179.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 read mod 2 are:
%e 1,
%e 0, 1, 0,
%e 1, 1, 1, 1, 1,
%e 0, 0, 0, 1, 0, 0, 0,
%e 1, 0, 0, 1, 1, 1, 0, 0, 1,
%e 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0,
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
%e 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0,
%e 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1,
%e 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0,
%e ...
%p A267179 := proc(n,k)
%p A267177(n,k) mod 2 ;
%p end proc:
%p seq(seq(A267179(n,k),k=1..2*n-1),n=1..10) ; # _R. J. Mathar_, May 08 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 A267179[n_, k_] := If[k <= n, A072030[n, k], A072030[2n-k, n]]~Mod~2;
%t Table[A267179[n, k], {n, 1, 10}, {k, 1, 2n-1}] // Flatten (* _Jean-François Alcover_, Apr 23 2023, after _R. J. Mathar_ *)
%o (PARI)
%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%2, ", "); );
%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%2, ", "); );
%o print(); ); }
%o tabl(12)
%Y This is A267177 read mod 2.
%Y Cf. A072030.
%K nonn,tabf
%O 1
%A _N. J. A. Sloane_, Jan 14 2016