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A077949
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Expansion of 1/(1-x-2*x^3).
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15
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1, 1, 1, 3, 5, 7, 13, 23, 37, 63, 109, 183, 309, 527, 893, 1511, 2565, 4351, 7373, 12503, 21205, 35951, 60957, 103367, 175269, 297183, 503917, 854455, 1448821, 2456655, 4165565, 7063207, 11976517, 20307647, 34434061, 58387095, 99002389, 167870511, 284644701
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OFFSET
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0,4
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COMMENTS
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Row sums of the Riordan array (1, x*(1+2*x^2)). - Paul Barry, Jan 12 2006
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
Number of compositions of n into parts 1 and two sorts of parts 2. - Joerg Arndt, Aug 29 2013
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
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
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LINKS
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FORMULA
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a(n) = Sum_{k=0..floor(n/2)} C(n-2k, k)*2^k. - Paul Barry, Nov 18 2003
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
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
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
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MAPLE
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a:= n-> (<<1|1|0>, <0|0|1>, <2|0|0>>^n)[1, 1]:
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MATHEMATICA
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PROG
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(Magma) [n le 3 select 1 else Self(n-1)+2*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Mar 13 2014
(Sage) (1/(1-x-2*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jun 22 2019
(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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