A066709 Triangle T(r,c) of winning binary "same game" templates.

1, 0, 1, 1, 2, 1, 0, 2, 4, 1, 1, 5, 8, 5, 1, 0, 3, 14, 15, 6, 1, 1, 9, 25, 32, 21, 7, 1, 0, 4, 32, 62, 56, 28, 8, 1, 1, 14, 56, 109, 122, 84, 36, 9, 1, 0, 5, 60, 170, 242, 210, 120, 45, 10, 1, 1, 20, 105, 275, 436, 457, 330, 165, 55, 11, 1, 0, 6, 100, 375, 732, 912, 792, 495, 220, 66, 12, 1

Offset: 1

Author

Frank Ellermann, Dec 31 2001

Keywords

• nonn
• tabl

Comments

T(r,c) is the number of winning templates with length r and minimum matching string length c; equivalently, ternary digits totaling r+c. For a definition and row sums 1,1,4,7,20, etc. see A066345. For antidiagonal sums 1,0,2,2,4,9, etc. see A066346.

Examples

			Rows:
1;
0,1;
1,2,1;
0,2,4,1;
1,5,8,5,1;
0,3,14,15,6,1; ...
a(17) = T(6,2) = 3 winning templates with length 6 and total 8 = 6+2: 211211, 121121, 112112.
A035615(6) = 2*( 1*1+0*1+1*3+1*1+2*2+1*1+1*1+0*1+2*1+1*1 ) = 2*13 = 26.
		

Links

• Sean A. Irvine, Java program (github)

Crossrefs

Cf. A035615, A066345, A066346, A007318.

Formula

A035615(n) = 2 * Sum_{r=1..n-1, c=1..min(r,n-r)} T(r,c) * P(n-r,c) where P(n-r,c) = C(n-r-1,c-1) = (n-r-1)!/((n-r-c-2)!*(c-1)!).

Extensions

More terms from Sean A. Irvine, Nov 03 2023