A003948 Expansion of (1+x)/(1-5*x).

1, 6, 30, 150, 750, 3750, 18750, 93750, 468750, 2343750, 11718750, 58593750, 292968750, 1464843750, 7324218750, 36621093750, 183105468750, 915527343750, 4577636718750, 22888183593750, 114440917968750, 572204589843750, 2861022949218750, 14305114746093750

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

Links

T. D. Noe, Table of n, a(n) for n = 0..200

F. Faase, Counting Hamiltonian cycles in product graphs

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 to divisibility sequences

Index entries for linear recurrences with constant coefficients, signature (5).

Index entries for sequences related to trees

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

Cf. A003946, A003947, A003949, A003950, A003952, A003954, A029653.

Sequence in context: A170591 A170639 A170687 * A105488 A252699 A054117

Adjacent sequences: A003945 A003946 A003947 * A003949 A003950 A003951

Keyword

nonn,easy,nice,walk

Author

N. J. A. Sloane

Extensions

Definition corrected by Frans J. Faase, Feb 07 2009

Edited by N. J. A. Sloane, Dec 04 2009

Status

approved