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%I #62 Oct 12 2022 04:45:09
%S 1,1,1,4,7,10,22,43,73,139,268,487,904,1708,3169,5881,11005,20512,
%T 38155,71170,132706,247171,460681,858799,1600312,2982355,5558752,
%U 10359688,19306753,35983009,67062073,124982332,232931359,434117578,809064574
%N 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.
%C 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.
%C 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
%C 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
%C 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
%H Vincenzo Librandi, <a href="/A084386/b084386.txt">Table of n, a(n) for n = 0..1000</a>
%H D. Birmajer, J. B. Gil, and M. D. Weiner, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Gil/gil6.html">On the Enumeration of Restricted Words over a Finite Alphabet </a>, J. Int. Seq. 19 (2016) # 16.1.3, Example 9
%H Merrill Jensen, <a href="http://www.mninter.net/~mpjensen/njas.htm">Generating Functions</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,3).
%F a(n) = a(n-1) + 3*a(n-3). a(n) = A052900(n+3)/3.
%F G.f.: 1/(1-x-3*x^3).
%F a(n) = Sum_{k=0..floor(n/2)} binomial(n-2*k, k)*3^k. - _Paul Barry_, Nov 18 2003
%F 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
%F Starting (1 + x + 4*x^2 + ...), is the INVERT transform of (1 + 3*x^2). - _Gary W. Adamson_, Mar 27 2017
%F 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
%p seq(add(binomial(n-2*k,k)*3^k,k=0..floor(n/3)),n=0..34); # _Zerinvary Lajos_, Apr 03 2007
%t a[0]=a[1]=a[2]=1; a[n_] := a[n]=a[n-1]+3a[n-3]; Table[a[n], {n, 0, 34}]
%t LinearRecurrence[{1, 0, 3}, {1, 1, 1}, 37] (* _Robert G. Wilson v_, Jul 12 2014 *)
%o (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
%o (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
%Y Partial sums of A052900. Also A052900/3.
%Y Cf. A000930, A001045, A006130.
%K easy,nonn
%O 0,4
%A Merrill Jensen (mpjensen(AT)mninter.net), Jun 23 2003
%E Edited by _Dean Hickerson_, Jun 24 2003
%E Recurrence appended to the name by _Antti Karttunen_, Mar 28 2017
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