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URL: https://oeis.org/A357743

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A357743
Square array A(n, k), n, k >= 0, read by antidiagonals: A(0, 0) = 0, A(0, 1) = A(1, 0) = 1, for n, k >= 0, A(2*n, 2*k) = A(n, k), A(2*n, 2*k+1) = A(n, k) + A(n, k+1), A(2*n+1, 2*k) = A(n, k) + A(n+1, k), A(2*n+1, 2*k+1) = A(n, k+1) + A(n+1, k).
2
0, 1, 1, 1, 2, 1, 2, 3, 3, 2, 1, 3, 2, 3, 1, 3, 4, 5, 5, 4, 3, 2, 5, 3, 6, 3, 5, 2, 3, 5, 6, 5, 5, 6, 5, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 4, 5, 7, 8, 7, 7, 8, 7, 5, 4, 3, 7, 4, 9, 5, 10, 5, 9, 4, 7, 3, 5, 8, 9, 7, 8, 11, 11, 8, 7, 9, 8, 5, 2, 7, 5, 8, 3, 9, 6, 9, 3, 8, 5, 7, 2
OFFSET
0,5
COMMENTS
This sequence is closely related to A002487 and A355855: we can build this sequence:
- by starting from an equilateral triangle with values 0, 1, 1:
0
/ \
1---1
- and repeatedly applying the following substitution:
t
/ \
t / \
/ \ --> t+u---t+v
u---v / \ / \
/ \ / \
u----u+v----v
The sequence reduced modulo an odd prime number presents rich nonperiodic patterns (see illustrations in Links section).
LINKS
Rémy Sigrist, Colored representation of the first 512 antidiagonals (where the color is function of A(n, k) mod 3)
Rémy Sigrist, Colored representation of the first 512 antidiagonals (where the color is function of A(n, k) mod 5)
FORMULA
A(n, k) = A(k, n).
A(n, 0) = A002487(n).
A(n, 1) = A007306(n+1) for any n > 0.
EXAMPLE
Array A(n, k) begins:
n\k | 0 1 2 3 4 5 6 7 8 9 10
----+---------------------------------------
0 | 0 1 1 2 1 3 2 3 1 4 3
1 | 1 2 3 3 4 5 5 4 5 7 8
2 | 1 3 2 5 3 6 3 7 4 9 5
3 | 2 3 5 6 5 5 8 9 7 8 11
4 | 1 4 3 5 2 7 5 8 3 9 6
5 | 3 5 6 5 7 10 11 9 8 11 11
6 | 2 5 3 8 5 11 6 11 5 10 5
7 | 3 4 7 9 8 9 11 10 7 7 12
8 | 1 5 4 7 3 8 5 7 2 9 7
9 | 4 7 9 8 9 11 10 7 9 14 17
10 | 3 8 5 11 6 11 5 12 7 17 10
.
The first antidiagonals are:
0
1 1
1 2 1
2 3 3 2
1 3 2 3 1
3 4 5 5 4 3
2 5 3 6 3 5 2
3 5 6 5 5 6 5 3
1 4 3 5 2 5 3 4 1
4 5 7 8 7 7 8 7 5 4
PROG
(PARI) A(n, k) = { if (n==0 && k==0, 0, n==1 && k==0, 1, n==0 && k==1, 1, n%2==0 && k%2==0, A(n/2, k/2), n%2==0, A(n/2, (k-1)/2) + A(n/2, (k+1)/2), k%2==0, A((n-1)/2, k/2) + A((n+1)/2, k/2), A((n+1)/2, (k-1)/2) + A((n-1)/2, (k+1)/2)); }
CROSSREFS
See A358871 for a similar sequence.
Sequence in context: A305715 A165014 A358871 * A058063 A393800 A232094
KEYWORD
nonn,tabl
AUTHOR
Rémy Sigrist, Nov 29 2022
STATUS
approved