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A006772 Sum of spans of 2n-step polygons on square lattice.
(Formerly M2971)
2
0, 1, 3, 14, 70, 370, 2028, 11452, 66172, 389416, 2326202, 14070268, 86010680, 530576780, 3298906810, 20653559846, 130099026600, 823979294284, 5244162058026, 33523117491920, 215150177410088, 1385839069134800, 8956173544332434, 58056703069399056, 377396656568011618, 2459614847765495754, 16068572108927106202 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

REFERENCES

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

LINKS

Table of n, a(n) for n=1..27.

A. J. Guttmann and I. G. Enting, The size and number of rings on the square lattice, J. Phys. A 21 (1988), L165-L172.

EXAMPLE

From Andrey Zabolotskiy, Nov 09 2018: (Start)

There are no 2-step polygons (conventionally).

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

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.

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

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

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

a 2 X 2 square, having span 2.

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

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

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

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

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

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

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

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

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

(End)

CROSSREFS

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

Sequence in context: A270598 A001579 A327871 * A320421 A009020 A186765

Adjacent sequences:  A006769 A006770 A006771 * A006773 A006774 A006775

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

Name corrected, more terms from Andrey Zabolotskiy, Nov 09 2018

STATUS

approved

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Last modified January 26 17:17 EST 2020. Contains 331280 sequences. (Running on oeis4.)