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%I #72 Sep 08 2022 08:45:08
%S 1,1,1,3,5,7,13,23,37,63,109,183,309,527,893,1511,2565,4351,7373,
%T 12503,21205,35951,60957,103367,175269,297183,503917,854455,1448821,
%U 2456655,4165565,7063207,11976517,20307647,34434061,58387095,99002389,167870511,284644701
%N Expansion of 1/(1-x-2*x^3).
%C Row sums of the Riordan array (1, x*(1+2*x^2)). - _Paul Barry_, Jan 12 2006
%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, 3*a(n-3) equals the number of 3-colored compositions of n with all parts >=3, such that no adjacent parts have the same color. - _Milan Janjic_, Nov 27 2011
%C Number of compositions of n into parts 1 and two sorts of parts 2. - _Joerg Arndt_, Aug 29 2013
%C a(n+2) equals the number of words of length n on alphabet {0,1,2}, having at least two zeros between every two successive nonzero letters. - _Milan Janjic_, Feb 07 2015
%C Number of pairs of rabbits when there are 2 pairs per litter and offspring reach parenthood after 3 gestation periods; a(n) = a(n-1) + 2*a(n-3), with a(0) = a(1) = a(2) = 1. - _Robert FERREOL_, Oct 27 2018
%H Vincenzo Librandi, <a href="/A077949/b077949.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,2).
%F a(n) = Sum_{k=0..floor(n/2)} C(n-2k, k)*2^k. - _Paul Barry_, Nov 18 2003
%F a(n) = Sum_{k=0..n} C(k, floor((n-k)/2))*2^((n-k)/2)*(1+(-1)^(n-k))/2. - _Paul Barry_, Jan 12 2006
%F a(n) = term (1,1) in the 3x3 matrix [1,1,0; 0,0,1; 2,0,0]^n. - _Alois P. Heinz_, Aug 16 2008
%F G.f.: Q(0)/2, where Q(k) = 1 + 1/(1 - x*(2*k+1 + 2*x^2)/( x*(2*k+2 + 2*x^2) + 1/Q(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Aug 29 2013
%p a:= n-> (<<1|1|0>, <0|0|1>, <2|0|0>>^n)[1, 1]:
%p seq(a(n), n=0..40); # _Alois P. Heinz_, Aug 16 2008
%t CoefficientList[Series[1/(1-x-2*x^3), {x, 0, 50}], x] (* _Jean-François Alcover_, Mar 11 2014 *)
%t LinearRecurrence[{1, 0, 2}, {1, 1, 1}, 50] (* _Robert G. Wilson v_, Jul 12 2014 *)
%o (PARI) Vec(1/(1-x-2*x^3)+O(x^50)) \\ _Charles R Greathouse IV_, Sep 23 2012
%o (Magma) [n le 3 select 1 else Self(n-1)+2*Self(n-3): n in [1..50]]; // _Vincenzo Librandi_, Mar 13 2014
%o (Sage) (1/(1-x-2*x^3)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Jun 22 2019
%o (GAP) a:=[1,1,1];; for n in [4..30] do a[n]:=a[n-1]+2*a[n-3]; od; a; # _G. C. Greubel_, Jun 22 2019
%Y Unsigned version of A077974. Cf. A003229.
%K nonn,easy
%O 0,4
%A _N. J. A. Sloane_, Nov 17 2002
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