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A027927 Number of plane regions after drawing (in general position) a convex n-gon and all its diagonals. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

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.

Dairyko, Michael; Tyner, Samantha; Pudwell, Lara; Wynn, Casey. 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

M. 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,2n-4), T given by A027926.

a(n) = 1 + binomial(n, 4) + binomial(n-1, 2) = (n^4 - 6n^3 + 23n^2 - 42n + 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)/3, a(n+1) - a(n) = A004006(n-1) for n > 2. - Waldemar Puszkarz, Jan 22 2016

EXAMPLE

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

MATHEMATICA

a=2; b=3; s=4; lst={0, 1, s}; Do[a+=n; b+=a; s+=b; AppendTo[lst, s], {n, 2, 4!, 1}]; lst+1 (* Vladimir Joseph Stephan Orlovsky, May 24 2009 *)

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

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

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Last modified January 22 11:56 EST 2019. Contains 319363 sequences. (Running on oeis4.)