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A015446 Generalized Fibonacci numbers: a(n) = a(n-1) + 10*a(n-2). 14
1, 1, 11, 21, 131, 341, 1651, 5061, 21571, 72181, 287891, 1009701, 3888611, 13985621, 52871731, 192727941, 721445251, 2648724661, 9863177171, 36350423781, 134982195491, 498486433301, 1848308388211, 6833172721221 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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+(4m+1)^1/2)^n)-(1-(4m+1)^1/2))^n)]/[(2^n)(4m+1)^1/2), a(n)/a(n-1) appears to converge to (1+sqrt(4m+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).

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

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 *)

PROG

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

(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

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

CROSSREFS

Cf. A015447, A015443.

Sequence in context: A116525 A094623 A034922 * A254208 A083177 A110466

Adjacent sequences:  A015443 A015444 A015445 * A015447 A015448 A015449

KEYWORD

nonn,easy

AUTHOR

Olivier Gérard

STATUS

approved

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Last modified February 21 04:43 EST 2018. Contains 299389 sequences. (Running on oeis4.)