A003951 Expansion of g.f.: (1+x)/(1-8*x).

1, 9, 72, 576, 4608, 36864, 294912, 2359296, 18874368, 150994944, 1207959552, 9663676416, 77309411328, 618475290624, 4947802324992, 39582418599936, 316659348799488, 2533274790395904, 20266198323167232, 162129586585337856, 1297036692682702848

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Coordination sequence for infinite tree with valency 9.

Binomial transform is {1, 10, 91, 820, 7381, ...}, see A002452. - Philippe Deléham, Jul 22 2005

a(n) equals the number of words of length n on alphabet {0,1,...,8} with no two adjacent letters identical. - Milan Janjic, Jan 31 2015 [Corrected by David Nacin, May 31 2017]

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 310
• Index to divisibility sequences
• Index entries for linear recurrences with constant coefficients, signature (8).
• Index entries for sequences related to trees

Crossrefs

Cf. A003945.

Programs

GAP

k:=9;; Concatenation([1], List([1..25], n-> k*(k-1)^(n-1) )); # G. C. Greubel, Sep 24 2019

Magma

[1] cat [9*8^(n-1): n in [1..25]]; // Vincenzo Librandi, Dec 11 2012

Maple

k:=9; seq(`if`(n=0, 1, k*(k-1)^(n-1)), n = 0..25); # modified by G. C. Greubel, Sep 24 2019

Mathematica

Join[{1}, 9*8^Range[0, 25]] (* Vladimir Joseph Stephan Orlovsky, Jul 11 2011 *)
CoefficientList[Series[(1+x)/(1-8*x), {x, 0, 25}], x] (* Vincenzo Librandi, Dec 10 2012 *)

PARI

a(n)=if(n,9*8^n/8,1) \\ Charles R Greathouse IV, Mar 22 2016

Sage

k=9; [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 = 7. - Philippe Deléham, Jul 10 2005

a(0) = 1; for n>0, a(n) = 9*8^(n-1). - Vincenzo Librandi, Nov 18 2010

a(0) = 1, a(1) = 9, a(n) = 8*a(n-1). - Vincenzo Librandi, Dec 10 2012

E.g.f.: (9*exp(8*x) -1)/8. - G. C. Greubel, Sep 24 2019

Extensions

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