OFFSET
0,2
COMMENTS
Coordination sequence for infinite tree with valency 6.
The n-th term of the coordination sequence of the infinite tree with valency 2m is the same as the number of reduced words of size n in the free group on m generators. In the five sequences A003946, A003948, A003950, A003952, A003954, m is 2, 3, 4, 5, 6. - Avi Peretz (njk(AT)netvision.net.il), Feb 23 2001 and Ola Veshta (olaveshta(AT)my-deja.com), Mar 30 2001
Hamiltonian path in S_4 X P_2n.
For n>=1, a(n+1) is equal to the number of functions f:{1,2,...,n+1}->{1,2,3,4,5,6} such that for fixed, different x_1, x_2,...,x_n in {1,2,...,n+1} and fixed y_1, y_2,...,y_n in {1,2,3,4,5,6} we have f(x_i)<>y_i, (i=1..n). - Milan Janjic, May 10 2007
For n>=1, a(n) equals the numbers of words of length n over the alphabet {0..5} with no two adjacent letters identical. - Milan Janjic, Jan 31 2015 [Corrected by David Nacin, May 30 2017]
a(n) equals the numbers of sequences of length n on {0,...,5} where no two adjacent terms differ by three. - David Nacin, May 30 2017
It appears that these are the only n>1 for which alpha(n)=2n, where alpha(n) is the entry point of n in the Fibonacci sequence, see A001177. - Philippe Schnoebelen, Apr 11 2024
LINKS
T. D. Noe, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 307
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
A. M. Nemirovsky et al., Marriage of exact enumeration and 1/d expansion methods: lattice model of dilute polymers, J. Statist. Phys., 67 (1992), 1083-1108.
Index entries for linear recurrences with constant coefficients, signature (5).
FORMULA
G.f.: (1+x)/(1-5*x).
a(n) = Sum_{k=0..n} A029653(n, k)*x^k for x = 4. - Philippe Deléham, Jul 10 2005
The Hankel transform of this sequence is [1,-6,0,0,0,0,0,0,0,0,...]. - Philippe Deléham, Nov 21 2007
a(n) = 6*5^(n-1) for n>0, a(0)=1. - Vincenzo Librandi, Nov 18 2010
G.f.: 2/x - 5 - 8/(x*U(0)) where U(k)= 1 + 2/(3^k - 3^k/(2 + 1 - 12*x*3^k/(6*x*3^k + 1/U(k+1)))) ; (continued fraction, 4-step). - Sergei N. Gladkovskii, Oct 30 2012
E.g.f.: (6*exp(5*x) - 1)/5. - Ilya Gutkovskiy, Dec 10 2016
Sum_{n>=0} 1/a(n) = 29/24. - Bernard Schott, Oct 25 2021
MAPLE
k := 6; if n = 0 then 1 else k*(k-1)^(n-1); fi;
MATHEMATICA
q = 6; Join[{a = 1}, Table[If[n != 0, a = q*a - a, a = q*a], {n, 0, 25}]] (* and *) Join[{1}, 6*5^Range[0, 25]] (* Vladimir Joseph Stephan Orlovsky, Jul 11 2011 *)
Join[{1}, NestList[5#&, 6, 30]] (* Harvey P. Dale, Dec 31 2013 *)
CoefficientList[Series[(1+x)/(1-5x), {x, 0, 30}], x] (* Michael De Vlieger, Dec 10 2016 *)
PROG
(PARI) Vec((1+x)/(1-5*x)+O(x^30)) \\ Charles R Greathouse IV, Nov 20 2012
(Magma) [1] cat [6*5^(n-1): n in [1..30]]; // G. C. Greubel, Sep 24 2019
(Sage) [1]+[6*5^(n-1) for n in (1..30)] # G. C. Greubel, Sep 24 2019
(GAP) Concatenation([1], List([1..30], n-> 6*5^(n-1) )); # G. C. Greubel, Sep 24 2019
CROSSREFS
KEYWORD
nonn,easy,nice,walk
AUTHOR
EXTENSIONS
Definition corrected by Frans J. Faase, Feb 07 2009
Edited by N. J. A. Sloane, Dec 04 2009
STATUS
approved