A005033 Number of nonequivalent dissections of a polygon into n quadrilaterals by nonintersecting diagonals rooted at a cell up to rotation. (Formerly M3921)

1, 1, 5, 22, 116, 612, 3399, 19228, 111041, 650325, 3856892, 23107896, 139672312, 850624376, 5214734547, 32154708216, 199292232035, 1240877862315, 7758138260565, 48685766617950, 306558216362064, 1936246229757840, 12263985131919300, 77880114240872112

Offset

1,3

Comments

Column F(r) in Table 2 of Harary link. - Robert A. Russell, Nov 22 2025

References

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

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

F. Harary, E. M. Palmer, and R. C. Read, On the cell-growth problem for arbitrary polygons, computer printout, circa 1974

F. Harary, E. M. Palmer, and R. C. Read, On the cell-growth problem for arbitrary polygons, Discr. Math. 11 (1975), 371-389.

Formula

G.f.: z * (G(z)^4 + G(z^2)^2 + 2*G(z^4)) / 4, where G(z)=1+z*G(z)^3 is the g.f. for A001764. - Robert A. Russell, Nov 22 2025

Mathematica

u[n_, k_, r_] := r*Binomial[(k-1)*n + r, n]/((k-1)*n + r);
T[n_, k_] := DivisorSum[GCD[n-1, k], EulerPhi[#]*u[(n-1)/#, k, k/#]&]/k;
a[n_] := T[n, 4];
Array[a, 24] (* Jean-François Alcover, Aug 20 2019, after Andrew Howroyd *)

Crossrefs

Column k=4 of A295222.

Cf. A001764.

Sequence in context: A153789 A213167 A355398 * A127618 A127619 A127620

Adjacent sequences: A005030 A005031 A005032 * A005034 A005035 A005036

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Sean A. Irvine, Mar 11 2016

Name edited by Andrew Howroyd, Nov 20 2017

Status

approved