A236939 Number T(n,k) of equivalence classes of ways of placing k 10 X 10 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=10, 0<=k<=floor(n/10)^2, read by rows.

1, 1, 1, 1, 1, 3, 1, 3, 1, 6, 1, 6, 1, 10, 1, 10, 1, 15, 1, 15, 1, 21, 36, 6, 1, 1, 21, 113, 80, 14, 1, 28, 261, 461, 174, 1, 28, 483, 1665, 1234, 1, 36, 819, 4725, 6124, 1, 36, 1266, 11193, 23259, 1, 45, 1878, 23646, 73204

Offset

10,6

Links

Table of n, a(n) for n=10..64.

Christopher Hunt Gribble, C++ program

Christopher Hunt Gribble, Example graphics

Formula

It appears that:

T(n,0) = 1, n>= 10

T(n,1) = (floor((n-10)/2)+1)*(floor((n-10)/2+2))/2, n >= 10

T(c+2*10,2) = A131474(c+1)*(10-1) + A000217(c+1)*floor(10^2/4) + A014409(c+2), 0 <= c < 10, c even

T(c+2*10,2) = A131474(c+1)*(10-1) + A000217(c+1)*floor((10-1)(10-3)/4) + A014409(c+2), 0 <= c < 10, c odd

T(c+2*10,3) = (c+1)(c+2)/2(2*A002623(c-1)*floor((10-c-1)/2) + A131941(c+1)*floor((10-c)/2)) + S(c+1,3c+2,3), 0 <= c < 10 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

A236936(26,3), c = 8

A236939(29,3), c = 9

Example

The first 17 rows of T(n,k) are:
.\ k  0     1     2     3     4
n
10    1     1
11    1     1
12    1     3
13    1     3
14    1     6
15    1     6
16    1    10
17    1    10
18    1    15
19    1    15
20    1    21    36     6     1
21    1    21   113    80    14
22    1    28   261   461   174
23    1    28   483  1665  1234
24    1    36   819  4725  6124
25    1    36  1266 11193 23259
26    1    45  1878 23646 73204
.
T(20,3) = 6 because the number of equivalence classes of ways of placing 3 10 X 10 square tiles in a 20 X 20 square under all symmetry operations of the square is 6.

Crossrefs

Cf. A054252, A236679, A236560, A236757, A236800, A236829, A236865, A236915, A236936

Sequence in context: A322865 A238694 A320221 * A236936 A236915 A236865

Adjacent sequences: A236936 A236937 A236938 * A236940 A236941 A236942

Keyword

tabf,nonn

Author

Christopher Hunt Gribble, Feb 01 2014

Status

approved