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 A143454 Expansion of 1/(x^k*(1-x-3*x^(k+1))) for k=3. 5
 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, 167572342 (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..50); MATHEMATICA a[n_] := Sum[3^j*Binomial[n-3*j+3, j], {j, 0, (n+3)/3}]; Table[a[n], {n, 0, 50}] (* Jean-François 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); /* 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 * A318774 A065810 A123837 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 July 18 19:00 EDT 2019. Contains 325144 sequences. (Running on oeis4.)