A236915 Number T(n,k) of equivalence classes of ways of placing k 8 X 8 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=8, 0<=k<=floor(n/8)^2, read by rows.
1, 1, 1, 1, 1, 3, 1, 3, 1, 6, 1, 6, 1, 10, 1, 10, 1, 15, 25, 5, 1, 1, 15, 79, 65, 14, 1, 21, 187, 377, 174, 1, 21, 351, 1365, 1234, 1, 28, 606, 3900, 6124, 1, 28, 948, 9282, 23259, 1, 36, 1426, 19726, 73204, 1, 36, 2026, 38046, 199436
Offset: 8
Examples
T(16,3) = 5 because the number of equivalence classes of ways of placing 3 8 X 8 square tiles in an 16 X 16 square under all symmetry operations of the square is 5. The portrayal of an example from each equivalence class is: ._____________________ _____________________ | | | | |__________| | | | | | | | | | | | | | . | . | | . | | | | | | | . | | | | | | | | | | | | | |__________|__________| |__________| | | | | | |__________| | | | | | | | | | | | | | . | | | . | | | | | | | | | | | | | | | | | | | | |__________|__________| |__________|__________| . ._____________________ _____________________ | | | | | | | |__________| | | | | | | | |__________| | . | | | . | | | | | | | | | | . | | | | | | | | | . | |__________| | |__________| | | | | | | | | |__________| | | | | | | | |__________| | . | | | . | | | | | | | | | | | | | | | | | | | | |__________|__________| |__________|__________| . ._____________________ | | | | | | | | | | . |__________| | | | | | | | | | |__________| . | | | | | | | | | | | . |__________| | | | | | | | | | |__________|__________|
Links
- Christopher Hunt Gribble, Rows n = 8..23, flattened
- Christopher Hunt Gribble, C++ program
Formula
It appears that:
T(n,0) = 1, n>= 8
T(n,1) = (floor((n-8)/2)+1)*(floor((n-8)/2+2))/2, n >= 8
T(c+2*8,2) = A131474(c+1)*(8-1) + A000217(c+1)*floor((8-1)(8-3)/4) + A014409(c+2), 0 <= c < 8, c odd
T(c+2*8,3) = (c+1)(c+2)/2(2*A002623(c-1)*floor((8-c-1)/2) + A131941(c+1)*floor((8-c)/2)) + S(c+1,3c+2,3), 0 <= c < 8 where
S(c+1,3c+2,3) =
A054252(2,3), c = 0
A236679(5,3), c = 1
A236560(8,3), c = 2
A236757(11,3), c = 3
A236800(14,3), c = 4
A236829(17,3), c = 5
A236865(20,3), c = 6
A236915(23,3), c = 7
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