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Triangle read by rows: T(j,k) is the number of distinct edge segments in a j X k rectangular grid.
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%I #60 Jul 18 2017 13:30:16

%S 1,2,1,3,3,2,3,3,4,2,4,4,5,5,3,4,4,5,5,6,3,5,5,6,6,7,7,4,5,5,6,6,7,7,

%T 8,4,6,6,7,7,8,8,9,9,5,6,6,7,7,8,8,9,9,10,5,7,7,8,8,9,9,10,10,11,11,6,

%U 7,7,8,8,9,9,10,10,11,11,12,6,8,8,9,9,10,10,11,11,12,12,13,13,7,8,8,9,9,10,10,11,11,12,12,13,13,14,7

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

%C This gives the number 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 = 1, a(2) = 1 X 2 = 2, a(3) = 2 X 2 = 1, a(4) = 1 X 3 = 3.

%C Here j = A002260(n), k = A002024(n), and n = A000217(k-1) + j, then a(n) = if j = k, ceiling(j/2), else ceiling(j/2) + ceiling(k/2).

%H Doug Bell, <a href="/A287618/b287618.txt">Table of n, a(n) for n = 1..11325</a>, Rows n = 1..150, flattened.

%e Triangle begins:

%e 1;

%e 2, 1;

%e 3, 3, 2;

%e 3, 3, 4, 2;

%e 4, 4, 5, 5, 3;

%e 4, 4, 5, 5, 6, 3;

%e 5, 5, 6, 6, 7, 7, 4;

%e ...

%e For n = 9, the a(9) = 4 distinct edge segments [A,B,C,D] for a 3 X 4 rectangular grid are:

%e + - - - - + + A B B A +

%e | | C C

%e | | --> D D

%e | | C C

%e + - - - - + + A B B A +.

%t Table[Ceiling[j/2] + Boole[j != k] Ceiling[k/2], {j, 14}, {k, j}] // Flatten (* _Michael De Vlieger_, Jun 09 2017 *)

%Y Cf. A002260, A002024, A000217.

%Y Cf. A287688 (number of distinct edge segment pairs).

%K nonn,tabl,hear

%O 1,2

%A _Doug Bell_, May 28 2017