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) = 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
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
New name from Len Smiley, Oct 19 2001
STATUS
approved