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A000939 Number of inequivalent n-gons.
(Formerly M1280 N0491)
8
1, 2, 4, 14, 54, 332, 2246, 18264, 164950, 1664354, 18423144, 222406776, 2905943328, 40865005494, 615376173184, 9880209206458, 168483518571798, 3041127561315224, 57926238289970076, 1161157777643184900, 24434798429947993054, 538583682082245127336 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,2

COMMENTS

Here two n-gons are said to be equivalent if they differ in starting point, orientation, or by a rotation (but not by a reflection - for that see A000940)

REFERENCES

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

LINKS

T. D. Noe, Table of n, a(n) for n=3..100

S. W. Golomb and L. R. Welch, On the enumeration of polygons, Amer. Math. Monthly, 67 (1960), 349-353.

S. W. Golomb and L. R. Welch, On the enumeration of polygons, Amer. Math. Monthly, 67 (1960), 349-353. [Annotated scanned copy]

A. Stoimenow, Enumeration of chord diagrams and an upper bound for Vassiliev invariants, J. Knot Theory Ramifications, 7 (1998), no. 1, 93-114.

FORMULA

For formula see Maple lines.

EXAMPLE

Possibilities for n-gons without distinguished vertex can be encoded as permutation classes of vertices, two permutations being equivalent if they can be obtained from each other by circular rotation, translation mod n or complement to n+1.

n=3: 123.

n=4: 1234, 1243.

n=5: 12345, 12354, 12453, 13524.

n=6: 123456, 123465, 123564, 123645, 123654, 124365, 124635, 124653, 125364, 125463, 125634, 126435, 126453, 135264.

MAPLE

with(numtheory); # for n odd: Ed:=proc(n) local t1, d; t1:=0; for d from 1 to n do if n mod d = 0 then t1:=t1+phi(n/d)^2*d!*(n/d)^d; fi; od: t1/(2*n^2); end;

# for n even: Ee:=proc(n) local t1, d; t1:=2^(n/2)*(n/2)*(n/2)!; for d from 1 to n do if n mod d = 0 then t1:=t1+phi(n/d)^2*d!*(n/d)^d; fi; od: t1/(2*n^2); end; A000939:=n-> if n mod 2 = 0 then Ee(n) else Ed(n); fi;

MATHEMATICA

a[n_] := (t = If[OddQ[n], 0, 2^(n/2)*(n/2)*(n/2)!];

Do[If[Mod[n, d]==0 , t = t+EulerPhi[n/d]^2*d!*(n/d)^d], {d, 1, n}]; t/(2*n^2)); a /@ Range[3, 24]

(* Jean-Fran├žois Alcover, May 19 2011, after Maple prog. *)

CROSSREFS

Cf. A000940. Bisections give A094154, A094155.

For star polygons see A231091.

Sequence in context: A183949 A131180 A047990 * A109154 A030853 A306150

Adjacent sequences:  A000936 A000937 A000938 * A000940 A000941 A000942

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 05 2004

STATUS

approved

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Last modified February 19 22:38 EST 2019. Contains 320328 sequences. (Running on oeis4.)