A003950 Expansion of g.f.: (1+x)/(1-7*x).

1, 8, 56, 392, 2744, 19208, 134456, 941192, 6588344, 46118408, 322828856, 2259801992, 15818613944, 110730297608, 775112083256, 5425784582792, 37980492079544, 265863444556808, 1861044111897656, 13027308783283592, 91191161482985144, 638338130380896008

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• walk
• easy

Comments

Coordination sequence for infinite tree with valency 8.

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.

For n>=1, a(n) equals the number of words of length n on the alphabet {0,1,...,7} with no two adjacent letters identical. - Milan Janjic, Jan 31 2015 [Corrected by David Nacin, May 31 2017]

a(n) is the number of octonary sequences of length n such that no two consecutive terms have distance 4. - David Nacin, May 31 2017

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 309
• 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 (7).
• Index entries for sequences related to trees

Crossrefs

Cf. A003948, A003949, A003951.

Programs

GAP

k:=8;; Concatenation([1], List([1..25], n-> k*(k-1)^(n-1) )); # G. C. Greubel, Sep 24 2019

Magma

[1] cat [8*7^(n-1): n in [1..25]]; // Vincenzo Librandi, Dec 11 2012

Maple

k:=8; seq(`if`(n=0, 1, k*(k-1)^(n-1)), n = 0..25); # modified by G. C. Greubel, Sep 24 2019

Mathematica

Join[{1}, 8*7^Range[0, 25]] (* Vladimir Joseph Stephan Orlovsky, Jul 11 2011 *)
CoefficientList[Series[(1+x)/(1-7*x), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 10 2012 *)

PARI

a(n)=if(n,8*7^(n-1),1) \\ Charles R Greathouse IV, Mar 22 2016

Sage

k=8; [1]+[k*(k-1)^(n-1) for n in (1..25)] # G. C. Greubel, Sep 24 2019

Formula

a(n) = Sum_{k=0..n} A029653(n, k)*x^k for x = 6. - Philippe Deléham, Jul 10 2005

From Philippe Deléham, Nov 21 2007: (Start)

a(n) = 8*7^(n-1) for n>=1, a(0)=1 .

G.f.: (1+x)/(1-7x).

The Hankel transform of this sequence is [1,-8,0,0,0,0,0,0,0,0,...]. (End)

a(0)=1, a(1)=8, a(n) = 7*a(n-1). - Vincenzo Librandi, Dec 10 2012

E.g.f.: (8*exp(7*x) - 1)/7. - G. C. Greubel, Sep 24 2019

Extensions

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