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A375874
Number of distinct n X n patterns in the squiral tiling.
1
1, 2, 14, 70, 126, 270, 438, 630, 790, 958, 1134, 1542, 1974, 2430, 2910, 3414, 3942, 4494, 5070, 5670, 6142, 6622, 7110, 7606, 8110, 8622, 9142, 9670, 10206, 11406, 12630, 13878, 15150, 16446, 17766, 19110, 20478, 21870, 23286, 24726, 26190, 27678, 29190
OFFSET
0,2
COMMENTS
The squiral tiling, can be obtained as the limit pattern under the binary block substitution 0 -> [[1,0,1],[0,0,0][1,0,1]] and 1 -> [[0,1,0],[1,1,1][0,1,0]], when starting with the seed 0.
REFERENCES
M. Baake, and U. Grimm, Aperiodic Order. Volume 1: A Mathematical Invitation, Encyclopedia of Mathematics and its Applications No. 149 Cambridge University Press, Cambridge (2013).
B. Grünbaum and F. C. Shephard, Tilings and Patterns, W.H. Freeman 1987, MR0857454.
LINKS
FORMULA
a(n) = (4 + 8*A - 8*B)*(n - 1)^2 + (12 * 3^A + 24 * 3^B) * (n - 1) - 18 * 9^A, for n>=4 where A = floor(log3(n-2)), B = floor(log3((n-2)/2)), and log3 is the logarithm in base 3.
For n>=2;
a(3*n-2) = 9*a(n),
a(9*n-7) = 5*a(3*n+1) - 16*a(3*n) + 20*a(3*n-1),
a(9*n-4) = - a(3*n+1) + 5*a(3*n) + 5*a(3*n-1),
a(9*n-1) = 2*a(3*n+1) + 8*a(3*n) - a(3*n-1),
a(3*n) = a(3*n-1) + 3*a(n+1) - 3*a(n).
EXAMPLE
a(1) = 2, since there are 2 different 1X1 patterns in the squiral tiling; namely 0 and 1.
a(2) = 14, since there are 14 different 2X2 patterns in the squiral tiling; namely all 16 2X2 binary matrices except [[0,0],[0,0]] and [[1,1],[1,1]].
MAPLE
a:= n-> `if`(n<3, [1, 2, 14][n+1], ((A, B)-> (4+8*A-8*B)*(n-1)^2+
(12*3^A+24*3^B)*(n-1)-18*9^A)(ilog[3](n-2), ilog[3]((n-2)/2))):
seq(a(n), n=0..42); # Alois P. Heinz, Sep 18 2024
PROG
(PARI) a(n)=if(n<4, [1, 2, 14, 70][n+1], my(A=logint(n-2, 3), B=logint((n-2)\2, 3)); (4 + 8*A - 8*B)*(n - 1)^2 + (12 * 3^A + 24 * 3^B) * (n - 1) - 18 * 9^A) \\ Andrew Howroyd, Sep 18 2024
(Python)
from sympy import integer_log
def A375874(n):
if n<4: return (1, 2, 14, 70)[n]
a, b = integer_log(n-2, 3)[0]+1, integer_log((n>>1)-1, 3)[0]+1
return (n-1)*((1+(a-b<<1))*(n-1)+((c:=3**a)+(3**b<<1))<<1)-c**2<<1 # Chai Wah Wu, Sep 18 2024
CROSSREFS
Sequence in context: A271235 A084770 A086243 * A258138 A206947 A203241
KEYWORD
nonn
AUTHOR
Johan Nilsson, Sep 01 2024
STATUS
approved