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A287618 Triangle read by rows: T(j,k) is the number of distinct edge segments in a j X k rectangular grid.

%I A287618 #60 Jul 18 2017 13:30:16
%S A287618 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 A287618 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 A287618 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 A287618 Triangle read by rows: T(j,k) is the number of distinct edge segments in a j X k rectangular grid.
%C A287618 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 A287618 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 A287618 Doug Bell, <a href="/A287618/b287618.txt">Table of n, a(n) for n = 1..11325</a>, Rows n = 1..150, flattened.
%e A287618 Triangle begins:
%e A287618   1;
%e A287618   2, 1;
%e A287618   3, 3, 2;
%e A287618   3, 3, 4, 2;
%e A287618   4, 4, 5, 5, 3;
%e A287618   4, 4, 5, 5, 6, 3;
%e A287618   5, 5, 6, 6, 7, 7, 4;
%e A287618 ...
%e A287618 For n = 9, the a(9) = 4 distinct edge segments [A,B,C,D] for a 3 X 4 rectangular grid are:
%e A287618   + - - - - +       + A B B A +
%e A287618   |         |       C         C
%e A287618   |         |  -->  D         D
%e A287618   |         |       C         C
%e A287618   + - - - - +       + A B B A +.
%t A287618 Table[Ceiling[j/2] + Boole[j != k] Ceiling[k/2], {j, 14}, {k, j}] // Flatten (* _Michael De Vlieger_, Jun 09 2017 *)
%Y A287618 Cf. A002260, A002024, A000217.
%Y A287618 Cf. A287688 (number of distinct edge segment pairs).
%K A287618 nonn,tabl,hear
%O A287618 1,2
%A A287618 _Doug Bell_, May 28 2017