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

%I A066709 #17 Nov 03 2023 17:59:36
%S A066709 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,
%T A066709 62,56,28,8,1,1,14,56,109,122,84,36,9,1,0,5,60,170,242,210,120,45,10,
%U A066709 1,1,20,105,275,436,457,330,165,55,11,1,0,6,100,375,732,912,792,495,220,66,12,1
%N A066709 Triangle T(r,c) of winning binary "same game" templates.
%C A066709 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.
%H A066709 Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a066/A066709.java">Java program</a> (github)
%F A066709 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)!).
%e A066709 Rows:
%e A066709 1;
%e A066709 0,1;
%e A066709 1,2,1;
%e A066709 0,2,4,1;
%e A066709 1,5,8,5,1;
%e A066709 0,3,14,15,6,1; ...
%e A066709 a(17) = T(6,2) = 3 winning templates with length 6 and total 8 = 6+2: 211211, 121121, 112112.
%e A066709 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.
%Y A066709 Cf. A035615, A066345, A066346, A007318.
%K A066709 nonn,tabl
%O A066709 1,5
%A A066709 _Frank Ellermann_, Dec 31 2001
%E A066709 More terms from _Sean A. Irvine_, Nov 03 2023