OFFSET
0
COMMENTS
Start with a single cell W_0 = [1], and for any k > 0, W_k is obtained by arranging (2*k+1)^2-1 copies of W_{k-1} around a square with the same dimensions as W_{k-1} consisting entirely of zeros.
In the original Wallis sieve, we split cells instead of gluing them together.
This array is self-similar and nonperiodic.
The average value after n iterations equals Prod_{k = 1..n} ((2*k+1)^2-1)/(2*k+1)^2 and tends to Pi/4.
LINKS
Rémy Sigrist, Table of n, a(n) for n = 0..10010
Rémy Sigrist, Illustration of first iterations
Rémy Sigrist, Illustration of the fourth iteration
Wikipedia, Wallis sieve
FORMULA
A(n, k) = A(k, n).
EXAMPLE
The array begins:
n\k | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
----+-------------------------------------------------
0 | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 | 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1
2 | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
3 | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
4 | 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1
5 | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
6 | 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1
7 | 1 0 1 1 0 1 0 0 0 1 0 1 1 0 1
8 | 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1
9 | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
10 | 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1
11 | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
12 | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
13 | 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1
14 | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
PROG
(PARI) A(n, k) = { for (i = 1, oo, if (n==0 && k==0, return (1), my (m = 2*i+1); if (n%m==i && k%m==i, return (0); ); n \= m; k \= m; ); ); }
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Rémy Sigrist, Apr 04 2026
STATUS
approved