A267177 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).

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, 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, 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

Offset: 1

Author

N. J. A. Sloane, Jan 14 2016

Keywords

• nonn
• tabf
• easy

Comments

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.

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.

Examples

			The array in A072030 begins:
  1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
  2, 1, 3, 2, 4, 3, 5, 4, 6, 5, ...
  3, 3, 1, 4, 4, 2, 5, 5, 3, 6, ...
  4, 2, 4, 1, 5, 3, 5, 2, 6, 4, ...
  5, 4, 4, 5, 1, 6, 5, 5, 6, 2, ...
  6, 3, 2, 3, 6, 1, 7, 4, 3, 4, ...
  7, 5, 5, 5, 5, 7, 1, 8, 6, 6, ...
  8, 4, 5, 2, 5, 4, 8, 1, 9, 5, ...
  9, 6, 3, 6, 6, 3, 6, 9, 1, 10, ...
  10, 5, 6, 4, 2, 4, 6, 5, 10, 1, ...
  ...
The successive bottom and right-hand borders are:
  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, 3, 6,
  7, 5, 5, 5, 5, 7, 1, 7, 5, 5, 5, 5, 7,
  ...
		

Links

• R. J. Mathar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A072030, A049834, A267178 (parity).

Programs

Maple

A267177 := proc(n,k)
    if k <= n then
        A072030(n,k) ;
    else
        A072030(2*n-k,n) ;
    end if;
end proc:
seq(seq(A267177(n,k),k=1..2*n-1),n=1..10) ; # R. J. Mathar, May 07 2016

Mathematica

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]];
A267177[n_, k_] := If[k <= n, A072030[n, k], A072030[2n-k, n]];
Table[A267177[n, k], {n, 1, 10}, {k, 1, 2n-1}] // Flatten (* Jean-François Alcover, Apr 23 2023, after R. J. Mathar *)

PARI

\\ Based on Michel Marcus's program for A049834.
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 += q; a = b; b = r); print1(s, ", "); );
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, ", "); );
print(); ); }
tabl(12)