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A006772 Sum of spans of 2n-step polygons on square lattice.

%I A006772 M2971 #19 Nov 09 2018 16:23:07
%S A006772 0,1,3,14,70,370,2028,11452,66172,389416,2326202,14070268,86010680,
%T A006772 530576780,3298906810,20653559846,130099026600,823979294284,
%U A006772 5244162058026,33523117491920,215150177410088,1385839069134800,8956173544332434,58056703069399056,377396656568011618,2459614847765495754,16068572108927106202
%N A006772 Sum of spans of 2n-step polygons on square lattice.
%D A006772 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006772 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 A006772 From _Andrey Zabolotskiy_, Nov 09 2018: (Start)
%e A006772 There are no 2-step polygons (conventionally).
%e A006772 For n=2, the only 4-step polygon is a 1 X 1 square having span 1, so a(2)=1.
%e A006772 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 A006772 For n=4, there are the following 8-step polygons:
%e A006772 a 3 X 1 stick which can be rotated 2 ways having spans 3 and 1;
%e A006772 an L-tromino which can be rotated 4 ways, all having span 2;
%e A006772 a 2 X 2 square, having span 2.
%e A006772 So a(4) = 3 + 1 + 4*2 + 2 = 14.
%e A006772 For n=5, there are the following 10-step polygons:
%e A006772 a 4 X 1 stick which can be rotated 2 ways having spans 4 and 1;
%e A006772 an L-tetromino which can be rotated 2 ways with span 2 and 2 more ways with span 3, plus reflections;
%e A006772 a T-tetromino which can be rotated 2 ways with span 2 and 2 more ways with span 3;
%e A006772 an S-tetromino which can be rotated 2 ways having spans 3 and 2, plus reflections;
%e A006772 a 3 X 2 rectangle which can be rotated 2 ways having spans 3 and 2;
%e A006772 a 3 X 2 rectangle without one of its angular squares having same counts as L-tetromino.
%e A006772 So a(5) = 4 + 1 + 2 * 2*2*(2+3) + 2*(2+3) + 2*(3+2) + 3 + 2 = 70.
%e A006772 (End)
%Y A006772 Cf. A002931, A006773, A302336, A302337, A232103.
%K A006772 nonn
%O A006772 1,3
%A A006772 _N. J. A. Sloane_
%E A006772 Name corrected, more terms from _Andrey Zabolotskiy_, Nov 09 2018