A001612 a(n) = a(n-1) + a(n-2) - 1 for n > 1, a(0)=3, a(1)=2.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• hear

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

Examples

			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
		

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, and Sukanta Das, On the reversibility of ECAs with fully asynchronous updating: the recurrence point of view, hal-01571847 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 and 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).

Crossrefs

Cf. A000032, A000071, A274017.

Programs

Haskell

a001612 n = a001612_list !! n
a001612_list = 3 : 2 : (map (subtract 1) $
   zipWith (+) a001612_list (tail a001612_list))
-- Reinhard Zumkeller, May 26 2013

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 *)

PARI

a(n)=fibonacci(n+1)+fibonacci(n-1)+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

Extensions

Additional comments from Michael Somos, Jun 01 2000