A006490 a(1) = 1, a(2) = 0; for n > 2, a(n) = n*Fibonacci(n-2) (with the convention Fibonacci(0)=0, Fibonacci(1)=1).

1, 0, 3, 4, 10, 18, 35, 64, 117, 210, 374, 660, 1157, 2016, 3495, 6032, 10370, 17766, 30343, 51680, 87801, 148830, 251758, 425064, 716425, 1205568, 2025675, 3399004, 5696122, 9534330, 15941099, 26625280, 44426877, 74062506, 123360230, 205303932, 341416205, 567353376

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Number of circular binary words of length n having exactly one occurrence of 00. Example: a(5)=10 because we have 00111, 10011, 11001, 11100, 01110, 00101, 10010, 01001, 10100 and 01010. Column 1 of A119458. - Emeric Deutsch, May 20 2006

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Ricardo Gómez Aíza, Symbolic dynamical scales: modes, orbitals, and transversals, arXiv:2009.02669 [math.DS], 2020.
• L. Carlitz and R. Scoville, Zero-one sequences and Fibonacci numbers, Fibonacci Quarterly, 15 (1977), 246-254.
• J. P. McSorley, Counting structures in the Moebius ladder, Discrete Math., 184 (1998), 137-164.
• 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
• Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Crossrefs

Cf. A000045, A119458.

Programs

Magma

[n*Fibonacci(n-2): n in [1..40]]; // Vincenzo Librandi, Aug 07 2017

Maple

with(combinat): a[1]:=1: a[2]:=0: for n from 3 to 40 do a[n]:=n*fibonacci(n-2) od: seq(a[n],n=1..40); # Emeric Deutsch, May 20 2006
A006490:=(1-2*z+2*z**2)/(z**2+z-1)**2; # conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

Table[Sum[Fibonacci[n - 1], {i, 0, n}], {n, 0, 34}] (* Zerinvary Lajos, Jul 12 2009 *)
CoefficientList[Series[(1 - 2 x + 2 x^2) / (1 - x - x^2)^2, {x, 0, 33}], x] (* or *) LinearRecurrence[{2, 1, -2, -1}, {1, 0, 3, 4}, 40] (* Vincenzo Librandi, Aug 07 2017 *)

PARI

a(n) = n*fibonacci(n-2); \\ Michel Marcus, Aug 07 2017

Formula

G.f.: x(1-2x+2x^2)/(1-x-x^2)^2. - Emeric Deutsch, May 20 2006

a(n) = 2*a(n-1)+a(n-2)-2*a(n-3)-a(n-4). - Vincenzo Librandi, Aug 07 2017

Extensions

Better definition from Ralf Stephan, Nov 18 2004

More terms from Emeric Deutsch, May 20 2006