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 A001612 a(n) = a(n-1) + a(n-2) - 1 for n > 1, a(0)=3, a(1)=2. (Formerly M0974 N0364) 9
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS 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. From Petros Hadjicostas, Jan 11 2017: (Start) 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 REFERENCES 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). LINKS T. D. Noe, Table of n, a(n) for n = 0..500 Nazim Fatès, Biswanath Sethi, Sukanta Das, On the reversibility of ECAs with fully asynchronous updating: the recurrence point of view, Preprint, 2017. Martin Griffiths, On a Matrix Arising from a Family of Iterated Self-Compositions, Journal of Integer Sequences, 18 (2015), #15.11.8. Fumio Hazama, Spectra of graphs attached to the space of melodies, Discr. Math., 311 (2011), 2368-2383. See Table 5.2. Martin Herschend, Peter Jorgensen, Classification of higher wide subcategories for higher Auslander algebras of type A, arXiv:2002.01778 [math.RT], 2020. Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 97. Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009. Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992 Eric Weisstein's World of Mathematics, Independent Vertex Set Eric Weisstein's World of Mathematics, Vertex Cover Eric Weisstein's World of Mathematics, Wheel Graph Index entries for linear recurrences with constant coefficients, signature (2,0,-1). FORMULA G.f.: (3-4*x)/((1-x)*(1-x-x^2)). a(n) = a(n-1) + a(n-2) - 1. a(n) = A000032(n) + 1. a(n) = A000071(n+3) - 2*A000071(n). - Petros Hadjicostas, Jan 11 2017 EXAMPLE 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 MAPLE 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 MATHEMATICA Join[{b=3}, a=0; Table[c=a+b-1; a=b; b=c, {n, 100}]] (* Vladimir Joseph Stephan Orlovsky, Mar 15 2011 *) Table[Fibonacci[n] + Fibonacci[n - 2] + 1, {n, 20}] (* Eric W. Weisstein, Mar 31 2017 *) LinearRecurrence[{2, 0, -1}, {3, 2, 4}, 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 *) PROG (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)) -- Reinhard Zumkeller, May 26 2013 CROSSREFS Cf. A000032, A000071, A274017. Sequence in context: A354731 A164287 A086962 * A275901 A305369 A097092 Adjacent sequences:  A001609 A001610 A001611 * A001613 A001614 A001615 KEYWORD nonn,easy,hear AUTHOR EXTENSIONS Additional comments from Michael Somos, Jun 01 2000 STATUS approved

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Last modified August 11 15:10 EDT 2022. Contains 356066 sequences. (Running on oeis4.)