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
KEYWORD
nonn
AUTHOR
EXTENSIONS
Better definition from Ralf Stephan, Nov 18 2004
More terms from Emeric Deutsch, May 20 2006
STATUS
approved