A027927 Number of plane regions after drawing (in general position) a convex n-gon and all its diagonals.

1, 2, 5, 12, 26, 51, 92, 155, 247, 376, 551, 782, 1080, 1457, 1926, 2501, 3197, 4030, 5017, 6176, 7526, 9087, 10880, 12927, 15251, 17876, 20827, 24130, 27812, 31901, 36426, 41417, 46905, 52922, 59501, 66676, 74482, 82955, 92132, 102051, 112751, 124272, 136655, 149942, 164176, 179401

Offset

2,2

Comments

For n>=1, a(n+1) is the number of Grassmannian permutations that avoid a pattern, sigma, where sigma is a pattern of size 5 with exactly one descent. - Jessica A. Tomasko, Nov 15 2022

Links

Vincenzo Librandi, Table of n, a(n) for n = 2..10000

Lapo Cioni and Luca Ferrari, Enumerative Results on the Schröder Pattern Poset, In: Dennunzio A., Formenti E., Manzoni L., Porreca A. (eds) Cellular Automata and Discrete Complex Systems, AUTOMATA 2017, Lecture Notes in Computer Science, vol 10248.

Michael Dairyko, Samantha Tyner, Lara Pudwell, and Casey Wynn, Non-contiguous pattern avoidance in binary trees, Electron. J. Combin. 19 (2012), no. 3, Paper 22, 21 pp. MR2967227. - From N. J. A. Sloane, Feb 01 2013

J. B. Gil and J. Tomasko, Restricted Grassmannian permutations, ECA 2:4 (2022) Article S4PP6.

Milan Janjic, Hessenberg Matrices and Integer Sequences , J. Int. Seq. 13 (2010) # 10.7.8.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

a(n) = T(n, 2*n-4), T given by A027926.

a(n) = 1 + binomial(n, 4) + binomial(n-1, 2) = (n^4 - 6*n^3 + 23*n^2 - 42*n + 48)/24.

G.f.: x^2*(1 -3*x +5*x^2 -3*x^3 +x^4)/(1-x)^5. - Colin Barker, Jan 31 2012

a(n) = (1/6)*A152950(n-1)*A152948(n). - Bruno Berselli, Jan 31 2012

a(n) = A000217(A000217(n-2)+2)/3, a(n+1) - a(n) = A004006(n-1) for n > 2. - Waldemar Puszkarz, Jan 22 2016 [Adjusted for offset by Peter Munn, Jan 10 2023]

a(n) = 1 + Sum {i=3..5} binomial(n-1, i-1). - Jessica A. Tomasko, Nov 15 2022

Example

a(2)=1 (segment traced twice has only exterior).

Maple

seq((n^4 -6*n^3 +23*n^2 -42*n +48)/24, n=2..50); # G. C. Greubel, Sep 06 2019

Mathematica

LinearRecurrence[{5, -10, 10, -5, 1 }, {1, 2, 5, 12, 26}, 50] (* Vincenzo Librandi, Feb 01 2012 *)
S[n_] :=n*(n+1)/2; Table[S[S[n]+2]/3, {n, 0, 50}] (* Waldemar Puszkarz, Jan 22 2016 *)

Prog

(PARI) a(n)=n*(n^3-6*n^2+23*n-42)/24+2 \\ Charles R Greathouse IV, Jan 31 2012
(Magma) [(n^4 -6*n^3 +23*n^2 -42*n +48)/24: n in [2..50]]; // G. C. Greubel, Sep 06 2019
(Sage) [(n^4 -6*n^3 +23*n^2 -42*n +48)/24 for n in (2..50)] # G. C. Greubel, Sep 06 2019
(GAP) List([2..50], n-> (n^4 -6*n^3 +23*n^2 -42*n +48)/24); # G. C. Greubel, Sep 06 2019

Crossrefs

Cf. A006522 (does not count exterior of n-gon).

Cf. A000045, A000217, A004006, A027926, A228074.

Sequence in context: A221720 A258099 A132977 * A221948 A116717 A116725

Adjacent sequences: A027924 A027925 A027926 * A027928 A027929 A027930

Keyword

nonn,easy

Author

Clark Kimberling

Extensions

New name from Len Smiley, Oct 19 2001

Status

approved