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Triangle read by rows: T(j,k) is the number of distinct edge segment pairs in a j X k rectangular grid.
2

%I #24 Jul 10 2017 04:28:17

%S 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 66,78,91,59,45,55,66,78,91,105,120,76,55,66,78,91,105,120,136,153,94,

%U 66,78,91,105,120,136,153,171,190,115,78,91,105,120,136,153,171,190,210,231,137

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

%C 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 Here j = A002260(n), k = A002024(n), and n = A000217(k-1) + j.

%C Where j != k, a(n) = A000217(j + k).

%C Where j = k, a(n) is approximately A236312(j-2); a(n) >= A236312(j-2).

%H Doug Bell, <a href="/A287688/b287688.txt">Table of n, a(n) for n = 1..11325</a>, rows n = 1..150, flattened.

%e Triangle begins:

%e 2;

%e 6, 7;

%e 10, 15, 13;

%e 15, 21, 28, 22;

%e 21, 28, 36, 45, 32;

%e 28, 36, 45, 55, 66, 45;

%e 36, 45, 55, 66, 78, 91, 59;

%e ...

%e For n = 3, the a(3) = 7 pairs of edge segments for a 2 X 2 rectangular grid are:

%e + - - + + * * + + * - + + * - + + * - + + * - + + * - + + * - +

%e | | --\ | | | * | | | | | | | | * |

%e | | --/ | | | | | * | | | | * | | |

%e + - - + + - - +, + - - +, + - - +, + - * +, + * - +, + - - +, + - - +.

%Y Cf. A002260, A002024, A000217, A236312. Distinct edge segments A287618.

%K nonn,tabl

%O 1,1

%A _Doug Bell_, May 29 2017