OFFSET
0,4
COMMENTS
This comment covers an infinite family of growth sequences, where a(n) = a(n-1) + k*a(n-m). k is number of pairs per litter and m is periods until adulthood. G.f. = 1/(1-x-k*x^m). For example, A000930 has k=1 and m=3 while A006130 has k=3 and m=2.
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 >= 3, 4*a(n-3) equals the number of 4-colored compositions of n with all parts >= 3, such that no adjacent parts have the same color. - Milan Janjic, Nov 27 2011
a(n+2) equals the number of words of length n on alphabet {0,1,2,3}, having at least two zeros between every two successive nonzero letters. - Milan Janjic, Feb 07 2015
Number of compositions of n into one sort of part 1 and three sorts of part 3 (see the g.f.). - Joerg Arndt, Feb 07 2015
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
Merrill Jensen, Generating Functions
Index entries for linear recurrences with constant coefficients, signature (1,0,3).
FORMULA
a(n) = a(n-1) + 3*a(n-3). a(n) = A052900(n+3)/3.
G.f.: 1/(1-x-3*x^3).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-2*k, k)*3^k. - Paul Barry, Nov 18 2003
G.f.: W(0)/2, where W(k) = 1 + 1/(1 - x*(1 + 3*x^2)/(x*(1 + 3*x^2) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 28 2013
Starting (1 + x + 4*x^2 + ...), is the INVERT transform of (1 + 3*x^2). - Gary W. Adamson, Mar 27 2017
a(m+n) = a(m)*a(n) + 3*a(m-1)*a(n-2) + 3*a(m-2)*a(n-1). - Michael Tulskikh, Jun 23 2020
MAPLE
seq(add(binomial(n-2*k, k)*3^k, k=0..floor(n/3)), n=0..34); # Zerinvary Lajos, Apr 03 2007
MATHEMATICA
a[0]=a[1]=a[2]=1; a[n_] := a[n]=a[n-1]+3a[n-3]; Table[a[n], {n, 0, 34}]
LinearRecurrence[{1, 0, 3}, {1, 1, 1}, 37] (* Robert G. Wilson v, Jul 12 2014 *)
PROG
(PARI) a(n)=([0, 1, 0; 0, 0, 1; 3, 0, 1]^n*[1; 1; 1])[1, 1] \\ Charles R Greathouse IV, Oct 03 2016
(Magma) I:=[1, 1, 1]; [n le 3 select I[n] else Self(n-1)+3*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Mar 28 2017
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Merrill Jensen (mpjensen(AT)mninter.net), Jun 23 2003
EXTENSIONS
Edited by Dean Hickerson, Jun 24 2003
Recurrence appended to the name by Antti Karttunen, Mar 28 2017
STATUS
approved