|
| |
|
|
A226290
|
|
Irregular triangle read by rows: T(n,k) is the number of binary pattern classes in the (3,n)-rectangular grid with k '1's and (3n-k) '0's: two patterns are in same class if one of them can be obtained by a reflection or 180-degree rotation of the other.
|
|
35
|
|
|
|
1, 1, 2, 2, 1, 1, 2, 6, 6, 6, 2, 1, 1, 4, 13, 27, 39, 39, 27, 13, 4, 1, 1, 4, 22, 60, 139, 208, 252, 208, 139, 60, 22, 4, 1, 1, 6, 34, 129, 371, 794, 1310, 1675, 1675, 1310, 794, 371, 129, 34, 6, 1, 1, 6, 48, 218, 813, 2196, 4767, 8070, 11139, 12300, 11139, 8070, 4767, 2196, 813, 218, 48, 6, 1
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
Sum of rows (see example) gives A225827.
By columns:
Also the number of equivalence classes of ways of placing k 1 X 1 tiles in an n X 3 rectangle under all symmetry operations of the rectangle. - Christopher Hunt Gribble, Apr 24 2015
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
n\k 0 1 2 3 4 5 6 7 8 9 10 11 12
0 1
1 1 2 2 1
2 1 2 6 6 6 2 1
3 1 4 13 27 39 39 27 13 4 1
4 1 4 22 60 139 208 252 208 139 60 22 4 1
5 1 6 34 129 371 794 1310 1675 1675 1310 794 371 129 34 6 1
6 1 6 48 218 813 2196 4767 8070 11139 12300 11139 8070 4767 2196 813 218 48 6 1
...
The length of row n is 3*n+1.
|
|
|
MATHEMATICA
|
T[n_, k_] := (Binomial[3n, k] + If[OddQ[n] || EvenQ[k], Binomial[Quotient[3 n, 2], Quotient[k, 2]], 0] + Sum[Binomial[n, k - 2i] Binomial[n, i] + Binomial[3 Mod[n, 2], k - 2i] Binomial[3 Quotient[n, 2], i], {i, 0, Quotient[k, 2]}])/4; Table[T[n, k], {n, 0, 6}, {k, 0, 3n}] // Flatten (* Jean-François Alcover, Oct 06 2017, after Andrew Howroyd *)
|
|
|
PROG
|
(PARI)
T(n, k)={(binomial(3*n, k) + if(n%2==1||k%2==0, binomial(3*n\2, k\2), 0) + sum(i=0, k\2, binomial(n, k-2*i) * binomial(1*n, i) + binomial(3*(n%2), k-2*i) * binomial(3*(n\2), i)))/4}
for(n=0, 6, for(k=0, 3*n, print1(T(n, k), ", ")); print) \\ Andrew Howroyd, May 30 2017
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,tabf
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
