A143454 Expansion of 1/(x^k*(1 - x - 3*x^(k+1))) for k=3.

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

Offset: 0

Author

Alois P. Heinz, Aug 16 2008

Keywords

• nonn
• easy

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
• D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, Example 9.
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,3).

Crossrefs

3rd column of A143461.

Programs

Magma

[n le 4 select 3*n-2 else Self(n-1) +3*Self(n-4): n in [1..51]]; // G. C. Greubel, May 08 2021

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 *)
LinearRecurrence[{1,0,0,3}, {1,4,7,10}, 41] (* G. C. Greubel, May 08 2021 *)

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

PARI

my(p=Mod('x,'x^4-'x^3-3)); a(n) = vecsum(Vec(lift(p^(n+3)))); \\ Kevin Ryde, May 11 2021

Sage

def a(n): return 3*n+1 if (n<4) else a(n-1) + 3*a(n-4)
[a(n) for n in (0..40)] # G. C. Greubel, May 08 2021

Formula

G.f.: (1 + 3*x + 3*x^2 + 3*x^3) / (1 - x - 3*x^4). - R. J. Mathar, Aug 04 2019

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