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A003229
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a(n) = a(n-1) + 2*a(n-3) with a(0)=a(1)=1, a(2)=3.
(Formerly M2419)
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23
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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
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OFFSET
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0,3
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COMMENTS
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Equals eigensequence of an infinite lower triangular matrix with 1's in the main diagonal, 0's in the subdiagonal and 2's in the subsubdiagonal (the triangle in the lower section of A155761). - Gary W. Adamson, Jan 28 2009
The operation in the comment of Jan 28 2009 is equivalent to the INVERT transform of (1, 0, 2, 0, 0, 0, ...). - Gary W. Adamson, Jan 21 2017
For n>=1, a(n) equals the number of ternary words of length n-1 having at least 2 zeros between every two successive nonzero letters. - Milan Janjic, Mar 09 2015
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REFERENCES
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D. E. Daykin and S. J. Tucker, Introduction to Dragon Curves. Unpublished, 1976. See links below for an earlier version.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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G.f.: (1+2*z^2)/(1-z-2*z^3). - Simon Plouffe in his 1992 dissertation
a(n) = hypergeom([-n/3,-(1+n)/3,(1-n)/3],[-n/2,-(1+n)/2],-27/2)) for n>=3. - Peter Luschny, Mar 09 2015
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MAPLE
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seq(add(binomial(n-2*k, k)*2^k, k=0..floor(n/3)), n=1..38); # Zerinvary Lajos, Apr 03 2007
with(combstruct): SeqSeqSeqL := [T, {T=Sequence(S), S=Sequence(U, card >= 1), U=Sequence(Z, card >=3)}, unlabeled]: seq(count(SeqSeqSeqL, size=n+4), n=0..35); # Zerinvary Lajos, Apr 04 2009
a := n -> `if`(n<3, [1, 1, 3][n+1], hypergeom([-n/3, -(1+n)/3, (1-n)/3], [-n/2, -(1+n)/2], -27/2)); seq(simplify(a(n)), n=0..35); # Peter Luschny, Mar 09 2015
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MATHEMATICA
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PROG
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(Magma) I:=[1, 1, 3]; [n le 3 select I[n] else Self(n-1)+2*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Jun 12 2012
(Haskell)
a003229 n = a003229_list !! n
a003229_list = 1 : 1 : 3 : zipWith (+)
(map (* 2) a003229_list) (drop 2 a003229_list)
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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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