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A094286 Number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 6 and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = 1, s(n) = 1. 1
1, 1, 2, 4, 9, 21, 51, 127, 323, 835, 2187, 5787, 15435, 41419, 111659, 302059, 819243, 2226219, 6058155, 16503211, 44991659, 122727595, 334914219, 914235051, 2496201387, 6816678571, 18617371307, 50851322539, 138903833259, 379443202731, 1036559854251 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
a(n) is the number of Motzkin n-paths of height <= 4. - Alois P. Heinz, Nov 24 2023
LINKS
S. Felsner and D. Heldt, Lattice Path Enumeration and Toeplitz Matrices, J. Int. Seq. 18 (2015) # 15.1.3.
Daniel Heldt, On the mixing time of the face flip-and up/down Markov chain for some families of graphs, Dissertation, Mathematik und Naturwissenschaften der Technischen Universitat Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften, 2016.
FORMULA
a(n) = (1/12)*(4 + 3*2^n + (1-sqrt(3))^n + (1+sqrt(3))^n).
a(n) = 1/3 + 2^(n-2) + A026150(n)/6.
G.f.: -x*(1-3*x+3*x^3) / ( (x-1)*(2*x-1)*(2*x^2+2*x-1) ). - R. J. Mathar, Dec 20 2011
MATHEMATICA
LinearRecurrence[{5, -6, -2, 4}, {1, 2, 4, 9}, 30] (* Harvey P. Dale, Feb 01 2012 *)
CROSSREFS
Cf. A001006.
Sequence in context: A005207 A257519 A257387 * A094287 A094288 A168051
KEYWORD
easy,nonn
AUTHOR
Herbert Kociemba, Jun 02 2004
EXTENSIONS
a(0)=1 prepended by Alois P. Heinz, Nov 24 2023
STATUS
approved

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Last modified May 22 06:24 EDT 2024. Contains 372743 sequences. (Running on oeis4.)