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A143454 Expansion of 1/(x^k*(1-x-3*x^(k+1))) for k=3. 4
1, 4, 7, 10, 13, 25, 46, 76, 115, 190, 328, 556, 901, 1471, 2455, 4123, 6826, 11239, 18604, 30973, 51451, 85168, 140980, 233899, 388252, 643756, 1066696, 1768393, 2933149, 4864417, 8064505, 13369684, 22169131, 36762382, 60955897, 101064949 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n) is also the number of length n quaternary words with at least 3 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>=7, 4*a(n-7) equals the number of 4-colored compositions of n with all parts >=4, such that no adjacent parts have the same color. - Milan Janjic, Nov 27 2011

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

FORMULA

G.f.: 1/(x^3*(1-x-3*x^4)).

a(n) = sum(j=0..(n+3)/3, 3^j*C(n-3*j+3,j)). - Vladimir Kruchinin, May 24 2011

MAPLE

a := proc(k::nonnegint) local n, i, j; if k=0 then unapply (4^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 3 else 0 fi)^(n+k))[1, 1], n) fi end(3): seq (a(n), n=0..47);

MATHEMATICA

a[n_] := Sum[3^j*Binomial[n-3*j+3, j], {j, 0, (n+3)/3}]; Table[a[n], {n, 0, 50}] (* Jean-Fran├žois Alcover, Feb 04 2014, after Vladimir Kruchinin *)

PROG

(Maxima) a(n):= sum(3^j*binomial(n-3*j+3, j), j, 0, (n+3)/3); [From Vladimir Kruchinin, May 24 2011]

(PARI) Vec(1/(x^3*(1-x-3*x^4))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

CROSSREFS

3rd column of A143461.

Sequence in context: A069212 A091290 A119256 * A065810 A123837 A125620

Adjacent sequences:  A143451 A143452 A143453 * A143455 A143456 A143457

KEYWORD

nonn,easy

AUTHOR

Alois P. Heinz, Aug 16 2008

STATUS

approved

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Last modified March 28 22:27 EDT 2017. Contains 284249 sequences.