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A052980
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Expansion of (1 - x)/(1 - 2*x - x^3).
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17
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1, 1, 2, 5, 11, 24, 53, 117, 258, 569, 1255, 2768, 6105, 13465, 29698, 65501, 144467, 318632, 702765, 1549997, 3418626, 7540017, 16630031, 36678688, 80897393, 178424817, 393528322, 867954037, 1914332891, 4222194104, 9312342245
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OFFSET
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0,3
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COMMENTS
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a(n) counts permutations of length n which embed into the (infinite) increasing oscillating sequence given by 4,1,6,3,8,5,...,2k+2,2k-1,...; these are also the permutations which avoid {321, 2341, 3412, 4123}. - Vincent Vatter, May 23 2008
a(n) is the top left entry of the n-th power of any of the 3X3 matrices [1, 1, 0; 1, 1, 1; 1, 0, 0] or [1, 1, 1; 1, 1, 0; 0, 1, 0] or [1, 1, 1; 0, 0, 1; 1, 0, 1] or [1, 0, 1; 1, 0, 0; 1, 1, 1]. - R. J. Mathar, Feb 03 2014
a(n) is the number of possible tilings of a 2 X n board, using dominos and L-shaped trominos. - Michael Tulskikh, Aug 21 2019
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
R. Brignall, N. Ruskuc and V. Vatter, Simple permutations: decidability and unavoidable substructures, Theoretical Computer Science 391 (2008), 150-163.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1053
D. Oudrar, Sur l'énumération de structures discrètes, une approche par la théorie des relations, Thesis (in French), arXiv:1604.05839 [math.CO], 2016.
D. Oudrar, M. Pouzet, Profile and hereditary classes of ordered relational structures, arXiv preprint arXiv:1409.1108 [math.CO], 2014 [The first version of this document erroneously gives the A-number as A005298]
V. Vatter, Small permutation classes, arXiv:0712.4006 [math.CO], 2007-2016.
Index entries for linear recurrences with constant coefficients, signature (2,0,1).
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FORMULA
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Recurrence: a(0)=1, a(1)=1, a(2)=2; thereafter a(n) = 2*a(n-1)+a(n-3).
Sum(1/59*(4+3*_alpha^2+17*_alpha)*_alpha^(-1-n), _alpha = RootOf(-1+2*_Z+_Z^3)).
a(n) = A008998(n) - A008998(n-1). - R. J. Mathar, Feb 04 2014
Let u1 = 2.20556943... denote the real root of x^3-2*x^2-1. There is an explicit constant c1 = 0.460719842... such that for n>0, a(n) = nearest integer to c1*u1^n. - N. J. A. Sloane, Nov 07 2016
a(2n) = a(n)^2 - a(n-1)^2 + (1/2)*(a(n+2) - a(n+1) - a(n))^2. - Greg Dresden and Michael Tulskikh, Aug 20 2019
a(n) = 2^(n-1) + Sum_{i=3..n}(2^(n-i)*a(i-3)). - Greg Dresden, Aug 27 2019
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MAPLE
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spec := [S, {S=Sequence(Prod(Union(Prod(Z, Z, Z), Z), Sequence(Z)))}, unlabeled ]: seq(combstruct[count ](spec, size=n), n=0..20);
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MATHEMATICA
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CoefficientList[Series[(1 - x)/(1 - 2 x - x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 05 2014 *)
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PROG
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(PARI) Vec((1-x)/(1-2*x-x^3)+O(x^99)) \\ Charles R Greathouse IV, Nov 20 2011
(MAGMA) I:=[1, 1, 2]; [n le 3 select I[n] else 2*Self(n-1)+Self(n-3): n in [1..40]]
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CROSSREFS
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See A110513 for another version.
Column k=2 of A219987.
Sequence in context: A286945 A111297 A077864 * A190512 A110513 A018115
Adjacent sequences: A052977 A052978 A052979 * A052981 A052982 A052983
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KEYWORD
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easy,nonn
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AUTHOR
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encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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STATUS
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approved
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