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A287688 Triangle read by rows: T(j,k) is the number of distinct edge segment pairs in a j X k rectangular grid.

%I A287688 #24 Jul 10 2017 04:28:17
%S A287688 2,6,7,10,15,13,15,21,28,22,21,28,36,45,32,28,36,45,55,66,45,36,45,55,
%T A287688 66,78,91,59,45,55,66,78,91,105,120,76,55,66,78,91,105,120,136,153,94,
%U A287688 66,78,91,105,120,136,153,171,190,115,78,91,105,120,136,153,171,190,210,231,137
%N A287688 Triangle read by rows: T(j,k) is the number of distinct edge segment pairs in a j X k rectangular grid.
%C A287688 This gives the number of pairs of edge segments that are distinct with respect to rotation and mirror images. Sequence is arranged so that j <= k (since 2 X 3 and 3 X 2 are equivalent grids), first by increasing j, then by increasing k: a(1) = 1 X 1 = 2, a(2) = 1 X 2 = 6, a(3) = 2 X 2 = 7, a(4) = 1 X 3 = 10.
%C A287688 Here j = A002260(n), k = A002024(n), and n = A000217(k-1) + j.
%C A287688 Where j != k, a(n) = A000217(j + k).
%C A287688 Where j = k, a(n) is approximately A236312(j-2); a(n) >= A236312(j-2).
%H A287688 Doug Bell, <a href="/A287688/b287688.txt">Table of n, a(n) for n = 1..11325</a>, rows n = 1..150, flattened.
%e A287688 Triangle begins:
%e A287688    2;
%e A287688    6,  7;
%e A287688   10, 15, 13;
%e A287688   15, 21, 28, 22;
%e A287688   21, 28, 36, 45, 32;
%e A287688   28, 36, 45, 55, 66, 45;
%e A287688   36, 45, 55, 66, 78, 91, 59;
%e A287688 ...
%e A287688 For n = 3, the a(3) = 7 pairs of edge segments for a 2 X 2 rectangular grid are:
%e A287688   + - - +     + * * +  + * - +  + * - +  + * - +  + * - +  + * - +  + * - +
%e A287688   |     | --\ |     |  |     *  |     |  |     |  |     |  |     |  *     |
%e A287688   |     | --/ |     |  |     |  |     *  |     |  |     |  *     |  |     |
%e A287688   + - - +     + - - +, + - - +, + - - +, + - * +, + * - +, + - - +, + - - +.
%Y A287688 Cf. A002260, A002024, A000217, A236312. Distinct edge segments A287618.
%K A287688 nonn,tabl
%O A287688 1,1
%A A287688 _Doug Bell_, May 29 2017