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

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

Author

N. J. A. Sloane

Keywords

• nonn

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, 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.

Crossrefs

Column k=4 of A295259.

Programs

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 *)

Extensions

More terms from Sean A. Irvine, Mar 11 2016

Name edited by Andrew Howroyd, Nov 20 2017