A015446 Generalized Fibonacci numbers: a(n) = a(n-1) + 10*a(n-2).

1, 1, 11, 21, 131, 341, 1651, 5061, 21571, 72181, 287891, 1009701, 3888611, 13985621, 52871731, 192727941, 721445251, 2648724661, 9863177171, 36350423781, 134982195491, 498486433301, 1848308388211, 6833172721221, 25316256603331, 93647983815541, 346810549848851

Offset: 0

Author

Olivier Gérard

Keywords

• nonn
• easy

Comments

The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n>=2, 11*a(n-2) equals the number of 11-colored compositions of n with all parts >=2, such that no adjacent parts have the same color. - Milan Janjic, Nov 26 2011

For a(n) = ((1+(4*m+1)^(1/2))^n - (1-(4*m+1)^(1/2))^n)/(2^n*(4*m+1)^(1/2)), a(n)/a(n-1) appears to converge to (1+sqrt(4*m+1))/2. Here with m = 10, the numbers in the sequence are congruent with those of the Fibonacci sequence modulo m-1 = 9. For example, F(8) = 21 (Fibonacci) corresponds to a(8) = 5061 (here) because 2+1 and 5+0+1+6 are congruent. - Maleval Francis, Nov 12 2013

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1, 10).

Crossrefs

Cf. A015447, A015443.

Programs

Magma

[ n eq 1 select 1 else n eq 2 select 1 else Self(n-1)+10*Self(n-2): n in [1..30] ]; // Vincenzo Librandi, Aug 23 2011

Mathematica

Table[MatrixPower[{{1,2},{5,0}},n][[1]][[1]],{n,0,44}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
CoefficientList[Series[1/(1-x-10*x^2), {x,0,50}], x] (* G. C. Greubel, Apr 30 2017 *)
LinearRecurrence[{1,10},{1,1},30] (* Harvey P. Dale, Dec 12 2018 *)

PARI

a(n)=([1,2;5,0]^n)[1,1] \\ Charles R Greathouse IV, Mar 09 2014

Sage

[lucas_number1(n,1,-10) for n in range(1, 25)] # Zerinvary Lajos, Apr 22 2009

Formula

a(n) = (((1+sqrt(41))/2)^(n+1) - ((1-sqrt(41))/2)^(n+1))/sqrt(41).

From Paul Barry, Sep 10 2005: (Start)

a(n) = Sum_{k=0..n} binomial((n+k)/2, k)*(1+(-1)^(n-k))*10^((n-k)/2)/2.

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*10^k. (End)

a(n) is the entry (M^n)1,1 where the matrix M = [1,2;5,0]. - _Simone Severini, Jun 22 2006

a(n) = Sum_{k=0..n} A109466(n,k)*(-10)^(n-k). - Philippe Deléham, Oct 26 2008

G.f.: 1/(1-x-10*x^2). - Colin Barker, Feb 03 2012

a(n) = (Sum_{k=1..n+1, k odd} C(n+1,k)*41^((k-1)/2))/2^n. - Vladimir Shevelev, Feb 05 2014