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). (Formerly M2362)

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

1,3

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).

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

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 *)

Prog

(Magma) [n*Fibonacci(n-2): n in [1..40]]; // Vincenzo Librandi, Aug 07 2017
(PARI) a(n) = n*fibonacci(n-2); \\ Michel Marcus, Aug 07 2017

Crossrefs

Cf. A000045, A119458.

Sequence in context: A280246 A338012 A350042 * A307856 A171160 A139797

Adjacent sequences: A006487 A006488 A006489 * A006491 A006492 A006493

Keyword

nonn,changed

Author

N. J. A. Sloane

Extensions

Better definition from Ralf Stephan, Nov 18 2004

More terms from Emeric Deutsch, May 20 2006

Status

approved