A005035 Number of nonequivalent dissections of a polygon into n quadrilaterals by nonintersecting diagonals rooted at a cell up to rotation and reflection. (Formerly M3471)

1, 1, 4, 13, 64, 315, 1727, 9658, 55657, 325390, 1929160, 11555172, 69840032, 425318971, 2607388905, 16077392564, 99646239355, 620439153165, 3879069845640, 24342884609625, 153279112388352, 968123122592340, 6131992590993204, 38940057166651848

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.: ((G(z^2)-1)/z + z^2*G(z^2)^4 + z*(G(z)^4 + G(z^2)^2 + 2*G(z^4))/4) / 2, 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);
F[n_, k_] := DivisorSum[GCD[n-1, k], EulerPhi[#]*u[(n-1)/#, k, k/#]&]/k;
T[n_, k_] := (F[n, k] + If[OddQ[k], If[OddQ[n], u[(n-1)/2, k, (k-1)/2], u[n/2-1, k, k-1]], If[OddQ[n], u[(n-1)/2, k, k/2+1], u[n/2-1, k, k]]])/2;
a[n_] := T[n, 4];
Array[a, 24] (* Jean-François Alcover, Jul 02 2018, after Andrew Howroyd *)

Crossrefs

Column k=4 of A295259.

Cf. A001764.

Sequence in context: A135312 A287145 A266096 * A340452 A222771 A052415

Adjacent sequences: A005032 A005033 A005034 * A005036 A005037 A005038

Keyword

nonn,changed

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