A267181 Array read by antidiagonals: T(i,j) (i>=0, j>=0) = number of steps to reach either top row or main diagonal using the steps (i,j)->(j,i) or (i,j)->(i,j-i).

0, 1, 0, 1, 0, 0, 1, 2, 1, 0, 1, 3, 0, 2, 0, 1, 4, 4, 3, 3, 0, 1, 5, 2, 0, 1, 4, 0, 1, 6, 5, 5, 4, 4, 5, 0, 1, 7, 3, 6, 0, 5, 2, 6, 0, 1, 8, 6, 2, 6, 5, 1, 5, 7, 0, 1, 9, 4, 6, 4, 0, 3, 5, 3, 8, 0, 1, 10, 7, 7, 7, 7, 6, 6, 6, 6, 9, 0, 1, 11, 5, 3, 2, 7, 0, 6, 1, 2, 4, 10, 0

Offset

0,8

Comments

We start at (i,j) and apply either (i,j) -> (j,i) if i>j or (i,j) -> (i,j-i) if j>i. T(i,j) is the minimal number of steps to reach either (0,k) or (k,k) for some k.

Somewhat analogous to the array in A072030 except that here the offset is different and we pay for transposition steps as well as subtraction steps.

Links

Table of n, a(n) for n=0..90.

Formula

Recurrence: T(0,k)=TR(k,k)=0; if i>j then T(i,j)=T(j,i)+1; if j>i then T(i,j)=T(i,j-i)+1.

For a > 1 and b,k > 0, T(ak,k) = a, T(ak+b,k) = T(b,k) + a + 2, T(k,ak) = a - 1, T(k,ak+b) = T(k,b) + a. - Charlie Neder, Feb 08 2019

Example

Array begins:
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...
1, 2, 0, 3, 1, 4, 2, 5, 3, 6, 4, 7, 5, ...
1, 3, 4, 0, 4, 5, 1, 5, 6, 2, 6, 7, 3, ...
1, 4, 2, 5, 0, 5, 3, 6, 1, 6, 4, 7, 2, ...
1, 5, 5, 6, 6, 0, 6, 6, 7, 7, 1, 7, 7, ...
1, 6, 3, 2, 4, 7, 0, 7, 4, 3, 5, 8, 1, ...
1, 7, 6, 6, 7, 7, 8, 0, 8, 7, 7, 8, 8, ...
1, 8, 4, 7, 2, 8, 5, 9, 0, 9, 5, 8, 3, ...
1, 9, 7, 3, 7, 8, 4, 8, 10, 0, 10, 8, 4, ...
1, 10, 5, 7, 5, 2, 6, 8, 6, 11, 0, 11, 6, ...
1, 11, 8, 8, 8, 8, 9, 9, 9, 9, 12, 0, 12, ...
1, 12, 6, 4, 3, 8, 2, 9, 4, 5, 7, 13, 0, ...
...
The first few antidiagonals are:
0,
1, 0,
1, 0, 0,
1, 2, 1, 0,
1, 3, 0, 2, 0,
1, 4, 4, 3, 3, 0,
1, 5, 2, 0, 1, 4, 0,
1, 6, 5, 5, 4, 4, 5, 0,
1, 7, 3, 6, 0, 5, 2, 6, 0,
1, 8, 6, 2, 6, 5, 1, 5, 7, 0,
1, 9, 4, 6, 4, 0, 3, 5, 3, 8, 0,
...

Maple

M:=12;
A:=Array(0..M, 0..M, 0);
for k from 0 to M do A[0, k]:=0; A[k, k]:=0; od:
# border number k
# col k, row n
for k from 1 to M do
for n from 1 to k-1 do A[n, k]:=A[n, k-n]+1; od:
# row k, col i
for i from k-1 by -1 to 0 do A[k, i]:=A[i, k]+1; od:
od:
for n from 0 to M do lprint([seq(A[n, k], k=0..M)]); od: # square array
for n from 0 to M do lprint([seq(A[n-j, j], j=0..n)]); od: # antidiagonals

Crossrefs

Cf. A072030.

For initial rows and columns see A267182-A267187.

For the array read mod 2, see A267188.

Sequence in context: A231119 A129558 A353414 * A131185 A307819 A286354

Adjacent sequences: A267178 A267179 A267180 * A267182 A267183 A267184

Keyword

nonn,tabl

Author

N. J. A. Sloane, Jan 16 2016

Status

approved