A084386 Number of pairs of rabbits when there are 3 pairs per litter and offspring reach parenthood after 3 gestation periods; a(n) = a(n-1) + 3*a(n-3), with a(0) = a(1) = a(2) = 1.

1, 1, 1, 4, 7, 10, 22, 43, 73, 139, 268, 487, 904, 1708, 3169, 5881, 11005, 20512, 38155, 71170, 132706, 247171, 460681, 858799, 1600312, 2982355, 5558752, 10359688, 19306753, 35983009, 67062073, 124982332, 232931359, 434117578, 809064574

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

Partial sums of A052900. Also A052900/3.

Cf. A000930, A001045, A006130.

Sequence in context: A227686 A161863 A102649 * A275176 A024726 A024948

Adjacent sequences: A084383 A084384 A084385 * A084387 A084388 A084389

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