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Sum of spans of 2n-step polygons on square lattice.
(Formerly M2971)
2

%I M2971 #19 Nov 09 2018 16:23:07

%S 0,1,3,14,70,370,2028,11452,66172,389416,2326202,14070268,86010680,

%T 530576780,3298906810,20653559846,130099026600,823979294284,

%U 5244162058026,33523117491920,215150177410088,1385839069134800,8956173544332434,58056703069399056,377396656568011618,2459614847765495754,16068572108927106202

%N Sum of spans of 2n-step polygons on square lattice.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H A. J. Guttmann and I. G. Enting, <a href="https://doi.org/10.1088/0305-4470/21/3/009">The size and number of rings on the square lattice</a>, J. Phys. A 21 (1988), L165-L172.

%e From _Andrey Zabolotskiy_, Nov 09 2018: (Start)

%e There are no 2-step polygons (conventionally).

%e For n=2, the only 4-step polygon is a 1 X 1 square having span 1, so a(2)=1.

%e For n=3, the only 6-step polygon is a 2 X 1 domino which can be rotated 2 ways having spans 2 and 1, so a(3) = 2+1 = 3.

%e For n=4, there are the following 8-step polygons:

%e a 3 X 1 stick which can be rotated 2 ways having spans 3 and 1;

%e an L-tromino which can be rotated 4 ways, all having span 2;

%e a 2 X 2 square, having span 2.

%e So a(4) = 3 + 1 + 4*2 + 2 = 14.

%e For n=5, there are the following 10-step polygons:

%e a 4 X 1 stick which can be rotated 2 ways having spans 4 and 1;

%e an L-tetromino which can be rotated 2 ways with span 2 and 2 more ways with span 3, plus reflections;

%e a T-tetromino which can be rotated 2 ways with span 2 and 2 more ways with span 3;

%e an S-tetromino which can be rotated 2 ways having spans 3 and 2, plus reflections;

%e a 3 X 2 rectangle which can be rotated 2 ways having spans 3 and 2;

%e a 3 X 2 rectangle without one of its angular squares having same counts as L-tetromino.

%e So a(5) = 4 + 1 + 2 * 2*2*(2+3) + 2*(2+3) + 2*(3+2) + 3 + 2 = 70.

%e (End)

%Y Cf. A002931, A006773, A302336, A302337, A232103.

%K nonn

%O 1,3

%A _N. J. A. Sloane_

%E Name corrected, more terms from _Andrey Zabolotskiy_, Nov 09 2018