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 *)
LinearRecurrence[{1, 10}, {1, 1}, 30] (* Harvey P. Dale, Dec 12 2018 *)
PROG
(Sage) [lucas_number1(n, 1, -10) for n in range(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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved