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

1,5

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.

Links

Table of n, a(n) for n=1..78.

Sean A. Irvine, Java program (github)

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)!).

Example

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.

Crossrefs

Cf. A035615, A066345, A066346, A007318.

Sequence in context: A320019 A141343 A256678 * A258051 A174026 A226075

Adjacent sequences: A066706 A066707 A066708 * A066710 A066711 A066712

Keyword

nonn,tabl

Author

Frank Ellermann, Dec 31 2001

Extensions

More terms from Sean A. Irvine, Nov 03 2023

Status

approved