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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).
8
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.
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
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Jan 16 2016
STATUS
approved