A015456 Generalized Fibonacci numbers.

1, 1, 11, 111, 1121, 11321, 114331, 1154631, 11660641, 117761041, 1189271051, 12010471551, 121293986561, 1224950337161, 12370797358171, 124932923918871, 1261700036546881, 12741933289387681, 128681032930423691, 1299552262593624591, 13124203658866669601

Offset: 0

Author

Olivier Gérard

Keywords

• nonn
• easy

Comments

For n>=1, row sums of triangle for numbers 10^k*C(m,k) with duplicated diagonals. - Vladimir Shevelev, Apr 13 2012

For n>=1, a(n) equals the numbers of words of length n-1 on alphabet {0,1,...,10} containing no subwords ii, (i=0,1,...,9). - Milan Janjic, Jan 31 2015

a(n) equals the number of sequences over the alphabet {0,1,...,9,10} such that no two consecutive terms have distance 6. - David Nacin, Jun 02 2017

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• M. Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (10,1).

Crossrefs

Row m=10 of A135597.

Programs

Magma

[n le 2 select 1 else 10*Self(n-1) + Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 08 2012

Mathematica

LinearRecurrence[{10, 1}, {1, 1}, 30] (* Vincenzo Librandi, Nov 08 2012 *)
CoefficientList[Series[(1-9*x)/(1-10*x-x^2), {x, 0, 50}], x] (* G. C. Greubel, Dec 19 2017 *)

PARI

x='x+O('x^30); Vec((1-9*x)/(1-10*x-x^2)) \\ G. C. Greubel, Dec 19 2017

Formula

a(n) = 10*a(n-1) + a(n-2).

a(n) = Sum_{k=0..n} 9^k*A055830(n,k). - Philippe Deléham, Oct 18 2006

G.f.: (1-9*x)/(1-10*x-x^2). - Philippe Deléham, Nov 20 2008

For n>=2, a(n) = F_(n)(10) + F_(n+1)(10), where F_n(x) is Fibonacci polynomial (cf.A049310): F_n(x) = Sum_{i=0,...,floor((n-1)/2)} C(n-i-1,i)*x^(n-2*i-1). - Vladimir Shevelev, Apr 13 2012