The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A220881 Number of nonequivalent dissections of an n-gon into n-3 polygons by nonintersecting diagonals up to rotation. 6
 1, 1, 4, 12, 43, 143, 504, 1768, 6310, 22610, 81752, 297160, 1086601, 3991995, 14732720, 54587280, 202997670, 757398510, 2834510744, 10637507400, 40023636310, 150946230006, 570534578704, 2160865067312, 8199711378716, 31170212479588, 118686578956272 (list; graph; refs; listen; history; text; internal format)
 OFFSET 4,3 COMMENTS This is almost identical to A003444, but has a different offset and a more precise definition. In other words, the number of almost-triangulations of an n-gon modulo the cyclic action. Equivalently, the number of edges of the (n-3)-dimensional associahedron modulo the cyclic action. The dissection will always be composed of one quadrilateral and n-4 triangles. - Andrew Howroyd, Nov 25 2017 Also number of necklaces of 2 colors with 2n-4 beads and n black ones. - Wouter Meeussen, Aug 03 2002 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS D. Bowman and A. Regev, Counting symmetry classes of dissections of a convex regular polygon, arXiv:1209.6270 [math.CO], 2012. P. Lisonek, Closed forms for the number of polygon dissections, Journal of Symbolic Computation 20 (1995), 595-601. C. R. Read, On general dissections of a polygon, Aequat. math. 18 (1978) 370-388. FORMULA a(n) = (1/(2n-4)) Sum_{d |(2n-4, n)} phi(d)*binomial((2n-4)/d, n/d) for n >= 4. - Wouter Meeussen, Aug 03 2002 MAPLE C:=n->binomial(2*n, n)/(n+1); T2:= proc(n) local t1; global C; t1 :=  (n-3)*C(n-2)/(2*n); if n mod 4 = 0 then t1:=t1+C(n/4-1)/2 fi; if n mod 2 = 0 then t1:=t1+C(n/2-1)/4 fi; t1; end; [seq(T2(n), n=4..40)]; MATHEMATICA c[n_] := Binomial[2*n, n]/(n+1); T2[n_] := Module[{t1}, t1 = (n-3)*c[n-2]/(2*n); If[Mod[n, 4] == 0, t1 = t1 + c[n/4-1]/2]; If[Mod[n, 2] == 0, t1 = t1 + c[n/2-1]/4]; t1]; Table[T2[n], {n, 4, 40}] (* Jean-François Alcover, Nov 23 2017, translated from Maple *) a[n_] := Sum[EulerPhi[d]*Binomial[(2n-4)/d, n/d], {d, Divisors[GCD[2n-4, n] ]}]/(2n-4); Array[a, 30, 4] (* Jean-François Alcover, Dec 02 2017, after Andrew Howroyd *) PROG (PARI) a(n) = if(n>=4, sumdiv(gcd(2*n-4, n), d, eulerphi(d)*binomial((2*n-4)/d, n/d))/(2*n-4)) \\ Andrew Howroyd, Nov 25 2017 CROSSREFS A diagonal of A295633. Cf. A003444, A003445. Cf. A003442, A003447, A003449. Sequence in context: A197659 A292287 A003444 * A149355 A149356 A149357 Adjacent sequences:  A220878 A220879 A220880 * A220882 A220883 A220884 KEYWORD nonn AUTHOR N. J. A. Sloane, Dec 28 2012 EXTENSIONS Name clarified by Andrew Howroyd, Nov 25 2017 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified July 12 03:28 EDT 2020. Contains 335658 sequences. (Running on oeis4.)