A005039 Number of nonequivalent dissections of a polygon into n pentagons by nonintersecting diagonals rooted at a cell up to rotation and reflection. (Formerly M3589)

1, 1, 4, 22, 147, 1074, 8216, 64798, 521900, 4272967, 35447724, 297308810, 2516830890, 21476307960, 184530904560, 1595190209002, 13863857007924, 121067796450692, 1061770618201680, 9347742325179544, 82584606893075739

Offset

1,3

Comments

Column f(r) in Table 3 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

Vaclav Kotesovec, Table of n, a(n) for n = 1..1000

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.: (1/10)*x*(u^5(x) + 4*u(x^5) + 5*u^2(x^2) + 5*x*u^4(x^2)) where u(x) is the g.f. for A002293. - Sean A. Irvine, Mar 12 2016

a(n) ~ 2^(8*n - 1/2) / (sqrt(Pi) * n^(3/2) * 3^(3*n + 5/2)). - Vaclav Kotesovec, Mar 13 2016

Mathematica

Rest[CoefficientList[Series[x*(HypergeometricPFQ[{1/4, 1/2, 3/4}, {2/3, 4/3}, (256/27)*x]^5 + 4*HypergeometricPFQ[{1/4, 1/2, 3/4}, {2/3, 4/3}, (256/27)*x^5] + 5*HypergeometricPFQ[{1/4, 1/2, 3/4}, {2/3, 4/3}, (256/27)*x^2]^2 + 5*x*HypergeometricPFQ[{1/4, 1/2, 3/4}, {2/3, 4/3}, (256/27)*x^2]^4)/10, {x, 0, 25}], x]] (* Vaclav Kotesovec, Mar 13 2016 *)

Crossrefs

Column k=5 of A295259.

Cf. A002293, A005037 (no mirror-image symmetries), A003446 (triangles), A005035 (quadrilaterals).

Sequence in context: A278396 A369617 A121394 * A199418 A112898 A368733

Adjacent sequences: A005036 A005037 A005038 * A005040 A005041 A005042

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Sean A. Irvine, Mar 12 2016

Name edited by Andrew Howroyd, Nov 20 2017

Status

approved