A358871 Square array A(n, k), n, k >= 0, read by antidiagonals: A(0, 0) = 0, A(0, 1) = A(1, 0) = 1, A(1, 1) = 2, 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+1, k+(1+(-1)^(n+k))/2) + A(n, k+(1-(-1)^(n+k))/2).
0, 1, 1, 1, 2, 1, 2, 3, 3, 2, 1, 3, 2, 3, 1, 3, 4, 5, 5, 4, 3, 2, 4, 3, 4, 3, 4, 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, 5, 4, 6, 5, 6, 5, 6, 4, 5, 3, 5, 7, 8, 7, 8, 9, 9, 8, 7, 8, 7, 5, 2, 6, 4, 7, 3, 7, 4, 7, 3, 7, 4, 6, 2
Offset: 0
Examples
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 4 5 4 5 5 7
2 | 1 3 2 5 3 6 3 7 4 8 4
3 | 2 3 5 4 5 5 8 6 7 7 10
4 | 1 4 3 5 2 7 5 8 3 9 6
5 | 3 4 6 5 7 6 9 7 8 8 11
6 | 2 5 3 8 5 9 4 9 5 10 5
7 | 3 4 7 6 8 7 9 6 7 7 12
8 | 1 5 4 7 3 8 5 7 2 9 7
9 | 4 5 8 7 9 8 10 7 9 8 13
10 | 3 7 4 10 6 11 5 12 7 13 6
.
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 4 3 4 3 4 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
Links
- Rémy Sigrist, Colored representation of the first 512 antidiagonals (where the color is function of A(n, k) mod 2)
- 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)
- Rémy Sigrist, Nonperiodic tilings related to Stern's diatomic series and based on tiles decorated with elements of Fp, arXiv:2301.06039 [math.CO], 2023.
Programs
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PARI
A(n,k) = { my (nn = n\2, kk=k\2); if (n<=1 && k<=1, n+k, n%2==0 && k%2==0, A(n/2,k/2), n%2==0, A(n/2,k\2)+A(n/2,k\2+1), k%2==0, A(n\2,k\2)+A(n\2+1,k\2), A(n\2+1,k\2+(1+(-1)^(n\2+k\2))/2) + A(n\2, k\2+(1-(-1)^(n\2+k\2))/2)); }
Comments