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A143447 Expansion of 1/(x^k*(1-x-2*x^(k+1))) for k=4. 3
1, 3, 5, 7, 9, 11, 17, 27, 41, 59, 81, 115, 169, 251, 369, 531, 761, 1099, 1601, 2339, 3401, 4923, 7121, 10323, 15001, 21803, 31649, 45891, 66537, 96539, 140145, 203443, 295225, 428299, 621377, 901667, 1308553, 1899003, 2755601, 3998355, 5801689, 8418795 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n) is also the number of length n ternary words with at least 4 0-digits between any other digits.

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>=9, 3*a(n-9) equals the number of 3-colored compositions of n with all parts >=5, such that no adjacent parts have the same color. - Milan Janjic, Nov 27 2011

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,2).

FORMULA

G.f.: 1/(x^4*(1-x-2*x^5)).

G.f.: Q(0)/(2*x^4) -1/x -1/x^2 -1/x^3 -1/x^4, where Q(k) = 1 + 1/(1 - x*(2*k+1 + 2*x^4)/( x*(2*k+2 + 2*x^4) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 29 2013

a(n) = 2n+1 if n<=5, else a(n) = a(n-1) + 2a(n-5). - Milan Janjic, Mar 09 2015

MAPLE

a:= proc(k::nonnegint) local n, i, j; if k=0 then unapply(3^n, n) else unapply((Matrix(k+1, (i, j)-> if (i=j-1) or j=1 and i=1 then 1 elif j=1 and i=k+1 then 2 else 0 fi)^(n+k))[1, 1], n) fi end(4): seq(a(n), n=0..54);

MATHEMATICA

Series[1/(1-x-2*x^5), {x, 0, 54}] // CoefficientList[#, x]& // Drop[#, 4]& (* Jean-François Alcover, Feb 13 2014 *)

CROSSREFS

4th column of A143453.

Sequence in context: A201644 A064076 A050842 * A152484 A071643 A039578

Adjacent sequences:  A143444 A143445 A143446 * A143448 A143449 A143450

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Aug 16 2008

STATUS

approved

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Last modified August 19 07:12 EDT 2017. Contains 290794 sequences.