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%I #75 Sep 08 2022 08:44:32
%S 1,8,56,392,2744,19208,134456,941192,6588344,46118408,322828856,
%T 2259801992,15818613944,110730297608,775112083256,5425784582792,
%U 37980492079544,265863444556808,1861044111897656,13027308783283592,91191161482985144,638338130380896008
%N Expansion of g.f.: (1+x)/(1-7*x).
%C Coordination sequence for infinite tree with valency 8.
%C 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.
%C 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]
%C 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
%H Vincenzo Librandi, <a href="/A003950/b003950.txt">Table of n, a(n) for n = 0..1000</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=309">Encyclopedia of Combinatorial Structures 309</a>
%H A. M. Nemirovsky et al., <a href="http://dx.doi.org/10.1007/BF01049010">Marriage of exact enumeration and 1/d expansion methods: lattice model of dilute polymers</a>, J. Statist. Phys., 67 (1992), 1083-1108.
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (7).
%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F a(n) = Sum_{k=0..n} A029653(n, k)*x^k for x = 6. - _Philippe Deléham_, Jul 10 2005
%F From _Philippe Deléham_, Nov 21 2007: (Start)
%F a(n) = 8*7^(n-1) for n>=1, a(0)=1 .
%F G.f.: (1+x)/(1-7x).
%F The Hankel transform of this sequence is [1,-8,0,0,0,0,0,0,0,0,...]. (End)
%F a(0)=1, a(1)=8, a(n) = 7*a(n-1). - _Vincenzo Librandi_, Dec 10 2012
%F E.g.f.: (8*exp(7*x) - 1)/7. - _G. C. Greubel_, Sep 24 2019
%p k:=8; seq(`if`(n=0, 1, k*(k-1)^(n-1)), n = 0..25); # modified by _G. C. Greubel_, Sep 24 2019
%t Join[{1}, 8*7^Range[0, 25]] (* _Vladimir Joseph Stephan Orlovsky_, Jul 11 2011 *)
%t CoefficientList[Series[(1+x)/(1-7*x), {x, 0, 30}], x] (* _Vincenzo Librandi_, Dec 10 2012 *)
%o (Magma) [1] cat [8*7^(n-1): n in [1..25]]; // _Vincenzo Librandi_, Dec 11 2012
%o (PARI) a(n)=if(n,8*7^(n-1),1) \\ _Charles R Greathouse IV_, Mar 22 2016
%o (Sage) k=8; [1]+[k*(k-1)^(n-1) for n in (1..25)] # _G. C. Greubel_, Sep 24 2019
%o (GAP) k:=8;; Concatenation([1], List([1..25], n-> k*(k-1)^(n-1) )); # _G. C. Greubel_, Sep 24 2019
%Y Cf. A003948, A003949, A003951.
%K nonn,walk,easy
%O 0,2
%A _N. J. A. Sloane_
%E Edited by _N. J. A. Sloane_, Dec 04 2009
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