The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A094287 Number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 7 and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = 1, s(n) = 1. 0

%I #26 Nov 24 2023 14:30:36

%S 1,1,2,4,9,21,51,127,323,835,2188,5798,15510,41822,113531,309937,

%T 850118,2340918,6466953,17913087,49726649,138287113,385126811,

%U 1073832695,2996974774,8370739326,23394528640,65415732100,182989086965,512046072481,1433197869570

%N Number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 7 and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = 1, s(n) = 1.

%C In general, a(n) = (2/m)*Sum_{k=1..m} sin(Pi*k/m)^2(1+2*cos(Pi*k/m))^n counts the (s(0), s(1), ..., s(n)) such that 0 < s(i) < m and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = 1, s(n) = 1. Here, m=7.

%C a(n) is the number of Motzkin n-paths of height <= 5. - _Alois P. Heinz_, Nov 24 2023

%H S. Felsner, D. Heldt, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Felsner/felsner2.html">Lattice Path Enumeration and Toeplitz Matrices</a>, J. Int. Seq. 18 (2015) # 15.1.3.

%H Daniel Heldt, <a href="http://dx.doi.org/10.14279/depositonce-5182">On the mixing time of the face flip-and up/down Markov chain for some families of graphs</a>, Dissertation, Mathematik und Naturwissenschaften der Technischen Universität Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften, 2016.

%F a(n) = (2/7)*Sum_{k=1..6} sin(Pi*k/7)^2(1+2*cos(Pi*k/7))^n.

%F Conjecture: a(n)= +6*a(n-1) -10*a(n-2) +9*a(n-4) -2*a(n-5) -a(n-6) with g.f. -x*(-1+4*x-2*x^2-5*x^3+2*x^4+x^5) / ( (x^3+3*x^2-4*x+1)*(x^3-x^2-2*x+1) ). - _R. J. Mathar_, Dec 20 2011

%t f[n_] := FullSimplify[ TrigToExp[(2/7)*Sum[ Sin[Pi*k/7]^2(1 + 2Cos[Pi*k/7])^n, {k, 1, 6}]]]; Table[ f[n], {n, 28}] (* _Robert G. Wilson v_, Jun 18 2004 *)

%Y Cf. A001006.

%K easy,nonn

%O 0,3

%A _Herbert Kociemba_, Jun 02 2004

%E a(0)=1 prepended by _Alois P. Heinz_, Nov 24 2023

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified June 18 21:09 EDT 2024. Contains 373487 sequences. (Running on oeis4.)