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A364823 Triangle read by rows: T(n,k) = number of possible positions for four connected discs in the game "Connect Four" played on a board with n columns and k rows, 4 <= k <= n.

%I A364823 #51 Oct 08 2023 09:24:37
%S A364823 10,17,28,24,39,54,31,50,69,88,38,61,84,107,130,45,72,99,126,153,180,
%T A364823 52,83,114,145,176,207,238,59,94,129,164,199,234,269,304,66,105,144,
%U A364823 183,222,261,300,339,378,73,116,159,202,245,288,331,374,417,460
%N A364823 Triangle read by rows: T(n,k) = number of possible positions for four connected discs in the game "Connect Four" played on a board with n columns and k rows, 4 <= k <= n.
%C A364823 In the game, all these positions can be reached. The most difficult thing is to connect four discs in the top row in the case of n=k. Here are examples for 4 X 4, 5 X 5 and 6 X 6:
%C A364823                                              .  b3 b12  b8 b11  .
%C A364823                       b3  b5  b8 b10  .      .  a3 a12  b7 a11  .
%C A364823   b2  b4  b8  b7      b2  a5  a8 a10  .      .  b2 b10  a7 a10  .
%C A364823   a2  a4  a8  b6      a2  b4  b7  b9  .      .  a2  a8  b6  b9  .
%C A364823   b1  b3  a7  a6      b1  a4  a7  a9  .      .  b1  a6  b5  a9  .
%C A364823   a1  a3  b5  a5      a1  a3  b6  a6  .      .  a1  b4  a4  a5  .
%C A364823 For n >= 7 any position in the top row can be reached by the following procedure. By repeating the following scheme, a tower of any height up to the second highest row can be built by placing discs alternately:
%C A364823   b4  b3  a4  a3
%C A364823   a1  a2  b1  b2
%C A364823 You can also build a separate tower where you are completely free with at least three discs. While one player places his four discs in the top row, the other moves to these reserve squares. Therefore, any position of four connected discs in the top row can be realized. Example 7 X 7:
%C A364823   .   a   a   a   a   .   .
%C A364823   .   b   b   a   a   .   .
%C A364823   .   a   a   b   b   .   .
%C A364823   .   b   b   a   a   .   .
%C A364823   .   a   a   b   b   .   b
%C A364823   .   b   b   a   a   .   b
%C A364823   .   a   a   b   b   .   b
%C A364823 For vertical positions there are many reserve squares in the other columns, for diagonal and horizontal positions other than in the top row you have additional reserve squares above three of the four discs to connect. For n > k you have further columns with more reserve squares.
%H A364823 Wikipedia, <a href="https://en.wikipedia.org/wiki/Connect_Four">Connect Four</a>.
%F A364823 T(n,k) = 4*k*n - 9*k - 9*n + 18, 4 <= k <= n, comprising k*(n-3) = k*n - 3*k horizontal positions, n*(k-3) = k*n - 3*n vertical positions, and 2*(n-3)*(k-3) = 2*k*n - 6*k - 6*n + 18 diagonal positions.
%F A364823 T(n,n) = 4*n^2 - 18*n + 18 = A059193(n-2).
%e A364823 The triangle T(n,k) begins:
%e A364823   n/k   4     5     6     7     8     9    10 ...
%e A364823    4:  10
%e A364823    5:  17    28
%e A364823    6:  24    39    54
%e A364823    7:  31    50    69    88
%e A364823    8:  38    61    84   107   130
%e A364823    9:  45    72    99   126   153   180
%e A364823   10:  52    83   114   145   176   207   238
%e A364823    .
%e A364823    .
%e A364823    .
%p A364823 A364823 := proc(n) local k; for k from 4 to n do return 4*k*n - 9*k - 9*n + 18; end do; end proc; seq(A364823(n), n = 4 .. 100);
%Y A364823 Cf. A013582, A090224, A212693, A059193.
%K A364823 nonn,tabl,easy
%O A364823 4,1
%A A364823 _Felix Huber_, Aug 09 2023