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%I #8 Aug 17 2024 09:16:44
%S 1,1,1,1,1,2,1,2,4,2,1,2,1,1,3,2,3,1,3,9,6,9,3,2,6,4,6,2,3,9,6,9,3,1,
%T 3,2,3,1,1,4,3,5,2,5,3,4,1,4,16,12,20,8,20,12,16,4,3,12,9,15,6,15,9,
%U 12,3,5,20,15,25,10,25,15,20,5,2,8,6,10,4,10,6,8,2
%N A family of squares S(m), m > 0, read by squares and then by rows; square S(1) is [1, 1; 1, 1]; for m > 0, square S(m+1) is obtained by replacing each subsquare [t, u; v, w] in S(m) by [t, t+u, u; t+v, t+u+v+w, u+w; v, v+w, w].
%C We apply the following substitutions to transform S(m) into S(m+1):
%C t----t+u----u
%C | | |
%C t--u | t+u |
%C | | --> t+v----+----u+w
%C v--w | v+w |
%C | | |
%C v----v+w----w
%C This sequence can be seen as a two-dimensional variant of A049456.
%C The base of T(m) corresponds to the m-th row of A049456.
%C As A355855, this sequence is related to nonperiodic tilings based on tiles decorated with elements of F_p for some odd prime number p; here we use square tiles, there triangular tiles.
%H Rémy Sigrist, <a href="/A375388/a375388.png">Colored representation of S(9) mod 3</a>
%H Rémy Sigrist, <a href="/A375388/a375388_1.png">Colored representation of S(9) mod 7</a>
%F S(m)(n, k) = A049456(m, n) * A049456(m, k).
%e S(1) is:
%e 1 1
%e 1 1
%e S(2) is:
%e 1 2 1
%e 2 4 2
%e 1 2 1
%e S(3) is:
%e 1 3 2 3 1
%e 3 9 6 9 3
%e 2 6 4 6 2
%e 3 9 6 9 3
%e 1 3 2 3 1
%e S(4) is:
%e 1 4 3 5 2 5 3 4 1
%e 4 16 12 20 8 20 12 16 4
%e 3 12 9 15 6 15 9 12 3
%e 5 20 15 25 10 25 15 20 5
%e 2 8 6 10 4 10 6 8 2
%e 5 20 15 25 10 25 15 20 5
%e 3 12 9 15 6 15 9 12 3
%e 4 16 12 20 8 20 12 16 4
%e 1 4 3 5 2 5 3 4 1
%o (PARI) S(n) = { matrix(2^(n-1)+1, 2^(n-1)+1, i,j, A002487(2^(n-1)-1+i) * A002487(2^(n-1)-1+j)); }
%Y Cf. A002487, A049456, A355855.
%K nonn,tabf,new
%O 1,6
%A _Rémy Sigrist_, Aug 13 2024
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