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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).
3
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
1,2
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.
LINKS
EXAMPLE
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,
...
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 *)
PROG
(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)
CROSSREFS
Cf. A072030, A049834, A267178 (parity).
Sequence in context: A210805 A303842 A057041 * A099567 A140530 A202191
KEYWORD
nonn,tabf,easy
AUTHOR
N. J. A. Sloane, Jan 14 2016
STATUS
approved