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A001612
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a(n) = a(n-1) + a(n-2) - 1 for n > 1, a(0)=3, a(1)=2.
(Formerly M0974 N0364)
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9
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3, 2, 4, 5, 8, 12, 19, 30, 48, 77, 124, 200, 323, 522, 844, 1365, 2208, 3572, 5779, 9350, 15128, 24477, 39604, 64080, 103683, 167762, 271444, 439205, 710648, 1149852, 1860499, 3010350, 4870848, 7881197, 12752044, 20633240, 33385283, 54018522
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OFFSET
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0,1
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COMMENTS
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a(n+3) = A^(n)B^(2)(1), n >= 0, with compositions of Wythoff's complementary A(n):=A000201(n) and B(n)=A001950(n) sequences. See the W. Lang link under A135817 for the Wythoff representation of numbers (with A as 1 and B as 0 and the argument 1 omitted). E.g., 5=`00`, 8=`100`, 12=`1100`, ..., in Wythoff code.
a(n) is the number of cyclic sequences consisting of zeros and ones that avoid the pattern 001 (or equivalently, the pattern 110) provided the positions of zeros and ones on a circle are fixed. This can easily be proved by considering that sequence A000071(n+3) is the number of binary zero-one words of length n that avoid the pattern 001 and that a(n) = A000071(n+3) - 2*A000071(n). (From the collection of all zero-one binary sequences that avoid 001 subtract those that start with 1 and end with 00 and those that start with 01 and end with 0.)
For n = 1,2, the number a(n) still gives the number of cyclic sequences consisting of zeros and ones that avoid the pattern 001 (provided the positions of zeros and ones on a circle are fixed) even if we assume that the sequence wraps around itself on the circle. For example, when 01 wraps around itself, it becomes 01010..., and it never contains the pattern 001. (End)
For n >= 3, a(n) is also the number of independent vertex sets and vertex covers in the wheel graph on n+1 nodes. - Eric W. Weisstein, Mar 31 2017
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REFERENCES
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N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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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Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 97.
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FORMULA
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G.f.: (3-4*x)/((1-x)*(1-x-x^2)).
a(n) = a(n-1) + a(n-2) - 1.
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EXAMPLE
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a(3) = 5 because the following cyclic sequences of length three avoid the pattern 001: 000, 011, 101, 110, 111. - Petros Hadjicostas, Jan 11 2017
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MAPLE
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A001612:=-(-2+3*z**2)/(z-1)/(z**2+z-1); # conjectured by Simon Plouffe in his 1992 dissertation; gives sequence except for the initial 3
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MATHEMATICA
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Table[Fibonacci[n] + Fibonacci[n - 2] + 1, {n, 20}] (* Eric W. Weisstein, Mar 31 2017 *)
CoefficientList[Series[(3 - 4 x)/(1 - 2 x + x^3), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 21 2017 *)
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PROG
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(PARI) a(n)=fibonacci(n+1)+fibonacci(n-1)+1
(Haskell)
a001612 n = a001612_list !! n
a001612_list = 3 : 2 : (map (subtract 1) $
zipWith (+) a001612_list (tail a001612_list))
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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