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

%I A027927 #99 May 07 2025 09:02:49
%S A027927 1,2,5,12,26,51,92,155,247,376,551,782,1080,1457,1926,2501,3197,4030,
%T A027927 5017,6176,7526,9087,10880,12927,15251,17876,20827,24130,27812,31901,
%U A027927 36426,41417,46905,52922,59501,66676,74482,82955,92132,102051,112751,124272,136655,149942,164176,179401
%N A027927 Number of plane regions after drawing (in general position) a convex n-gon and all its diagonals.
%C A027927 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
%H A027927 Vincenzo Librandi, <a href="/A027927/b027927.txt">Table of n, a(n) for n = 2..10000</a>
%H A027927 Lapo Cioni and Luca Ferrari, <a href="https://dx.doi.org/10.1007/978-3-319-58631-1_5">Enumerative Results on the Schröder Pattern Poset</a>, 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.
%H A027927 Michael Dairyko, Samantha Tyner, Lara Pudwell, and Casey Wynn, <a href="https://doi.org/10.37236/2099">Non-contiguous pattern avoidance in binary trees</a>, Electron. J. Combin. 19 (2012), no. 3, Paper 22, 21 pp. MR2967227. - From _N. J. A. Sloane_, Feb 01 2013
%H A027927 J. B. Gil and J. Tomasko, <a href="https://doi.org/10.54550/ECA2022V2S4PP6">Restricted Grassmannian permutations</a>, ECA 2:4 (2022) Article S4PP6.
%H A027927 Milan Janjić, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Janjic/janjic33.html">Hessenberg Matrices and Integer Sequences</a>, J. Int. Seq. 13 (2010) # 10.7.8.
%H A027927 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A027927 a(n) = T(n, 2*n-4), T given by A027926.
%F A027927 a(n) = 1 + binomial(n, 4) + binomial(n-1, 2) = (n^4 - 6*n^3 + 23*n^2 - 42*n + 48)/24.
%F A027927 G.f.: x^2*(1 -3*x +5*x^2 -3*x^3 +x^4)/(1-x)^5. - _Colin Barker_, Jan 31 2012
%F A027927 a(n) = (1/6)*A152950(n-1)*A152948(n). - _Bruno Berselli_, Jan 31 2012
%F A027927 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]
%F A027927 a(n) = 1 + Sum {i=3..5} binomial(n-1, i-1). - _Jessica A. Tomasko_, Nov 15 2022
%e A027927 a(2)=1 (segment traced twice has only exterior).
%p A027927 seq((n^4 -6*n^3 +23*n^2 -42*n +48)/24, n=2..50); # _G. C. Greubel_, Sep 06 2019
%t A027927 LinearRecurrence[{5,-10,10,-5,1 }, {1,2,5,12,26}, 50] (* _Vincenzo Librandi_, Feb 01 2012 *)
%t A027927 S[n_] :=n*(n+1)/2; Table[S[S[n]+2]/3, {n, 0, 50}] (* _Waldemar Puszkarz_, Jan 22 2016 *)
%o A027927 (PARI) a(n)=n*(n^3-6*n^2+23*n-42)/24+2 \\ _Charles R Greathouse IV_, Jan 31 2012
%o A027927 (Magma) [(n^4 -6*n^3 +23*n^2 -42*n +48)/24: n in [2..50]]; // _G. C. Greubel_, Sep 06 2019
%o A027927 (Sage) [(n^4 -6*n^3 +23*n^2 -42*n +48)/24 for n in (2..50)] # _G. C. Greubel_, Sep 06 2019
%o A027927 (GAP) List([2..50], n-> (n^4 -6*n^3 +23*n^2 -42*n +48)/24); # _G. C. Greubel_, Sep 06 2019
%Y A027927 Cf. A006522 (does not count exterior of n-gon).
%Y A027927 Cf. A000045, A000217, A004006, A027926, A228074.
%K A027927 nonn,easy
%O A027927 2,2
%A A027927 _Clark Kimberling_
%E A027927 New name from _Len Smiley_, Oct 19 2001