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A238552
Number T(n,k) of equivalence classes of ways of placing k 4 X 4 tiles in an n X 6 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=4, 0<=k<=floor(n/4), read by rows.
21
1, 2, 1, 2, 1, 4, 1, 4, 1, 6, 4, 1, 6, 9, 1, 8, 18, 1, 8, 28, 1, 10, 42, 10, 1, 10, 57, 28, 1, 12, 76, 76, 1, 12, 96, 140, 1, 14, 120, 254, 25, 1, 14, 145, 392, 107, 1, 16, 174, 600, 321, 1, 16, 204, 840, 731, 1, 18, 238, 1170, 1462, 70, 1, 18, 273, 1540, 2610, 366
OFFSET
4,2
LINKS
Christopher Hunt Gribble, C++ program
EXAMPLE
The first 14 rows of T(n,k) are:
.\ k 0 1 2 3 4
n
4 1 2
5 1 2
6 1 4
7 1 4
8 1 6 4
9 1 6 9
10 1 8 18
11 1 8 28
12 1 10 42 10
13 1 10 57 28
14 1 12 76 76
15 1 12 96 140
16 1 14 120 254 25
17 1 14 145 392 107
MATHEMATICA
T[n_, k_] := ((3^k + 1) Binomial[n - 3k, k] + Boole[EvenQ[k] || EvenQ[n]]*(3^Quotient[k + 1, 2] + 3^Quotient[k, 2]) * Binomial[(n - 3k - Mod[k, 2] - Mod[n, 2])/2, Quotient[k, 2]])/4; Table[T[n, k], {n, 4, 20}, {k, 0, Floor[n/4]}] // Flatten (* Jean-François Alcover, Oct 06 2017, after Andrew Howroyd *)
PROG
(C++) See Gribble link.
(PARI)
T(n, k)={((3^k+1)*binomial(n-3*k, k) + (k%2==0||n%2==0) * (3^((k+1)\2)+3^(k\2)) * binomial((n-3*k-(k%2)-(n%2))/2, k\2))/4}
for(n=4, 20, for(k=0, floor(n/4), print1(T(n, k), ", ")); print) \\ Andrew Howroyd, May 29 2017
KEYWORD
tabf,nonn
AUTHOR
EXTENSIONS
Terms corrected and xrefs updated by Christopher Hunt Gribble, Apr 27 2015
Terms a(28) and beyond from Andrew Howroyd, May 29 2017
STATUS
approved