A003952 Expansion of g.f.: (1+x)/(1-9*x).

1, 10, 90, 810, 7290, 65610, 590490, 5314410, 47829690, 430467210, 3874204890, 34867844010, 313810596090, 2824295364810, 25418658283290, 228767924549610, 2058911320946490, 18530201888518410, 166771816996665690, 1500946352969991210, 13508517176729920890

Offset: 0

Author

N. J. A. Sloane, Mar 15 1996

Keywords

• nonn
• easy

Comments

Coordination sequence for infinite tree with valency 10.

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

Except 1, all terms are in A033583. - Vincenzo Librandi, May 26 2014

For n>=1, a(n) equals the number of words of length n on alphabet {0,1,...,9} with no two adjacent letters identical. - Milan Janjic, Jan 31 2015 [Corrected by David Nacin, May 31 2017]

a(n) is the number of sequences over the alphabet {0,1,...,9} of length n such that no two consecutive terms have distance 5. - David Nacin, May 31 2017

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 311
• Index to divisibility sequences
• Index entries for linear recurrences with constant coefficients, signature (9).
• Index entries for sequences related to trees

Crossrefs

Cf. A003949, A003950, A003951.

Programs

GAP

k:=10;; Concatenation([1], List([1..20], n-> k*(k-1)^(n-1) )); # G. C. Greubel, Sep 24 2019

Magma

[(10*9^n-0^n)/9: n in [0..20] ]; // Vincenzo Librandi, Aug 19 2011

Maple

k:= 10; seq(`if`(n = 0, 1, k*(k-1)^(n-1)), n = 0..20); # modified by G. C. Greubel, Sep 24 2019

Mathematica

Join[{1}, 10*9^Range[0, 25]] (* Vladimir Joseph Stephan Orlovsky, Jul 11 2011 *)
Join[{1},NestList[9#&,10,20]] (* Harvey P. Dale, Sep 01 2021 *)

PARI

a(n)=10*9^n\9 \\ Charles R Greathouse IV, Sep 08 2011

Sage

k=10; [1]+[k*(k-1)^(n-1) for n in (1..20)] # G. C. Greubel, Sep 24 2019

Formula

a(n) = (10*9^n - 0^n)/9. Binomial transform is A000042. - Paul Barry, Jan 29 2004

G.f.: (1+x)/(1-9*x). - Philippe Deléham, Jan 31 2004

a(n) = Sum_{k=0..n} A029653(n, k)*x^k for x = 8. - Philippe Deléham, Jul 10 2005

The Hankel transform of this sequence is: [1,-10,0,0,0,0,0,0,0,...]. - Philippe Deléham, Nov 21 2007

E.g.f.: (10*exp(9*x) - 1)/9. - G. C. Greubel, Sep 24 2019

Extensions

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