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%I #71 Sep 08 2022 08:44:52
%S 0,2,7,20,54,143,376,986,2583,6764,17710,46367,121392,317810,832039,
%T 2178308,5702886,14930351,39088168,102334154,267914295,701408732,
%U 1836311902,4807526975,12586269024,32951280098,86267571271,225851433716
%N a(n) = Fibonacci(2*n+2) - 1.
%C Except for 0, numbers whose dual Zeckendorf representation (A104326) has the same number of 0's as 1's. - _Amiram Eldar_, Mar 22 2021
%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011. [Cached copy]
%H Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.
%H Clark Kimberling, <a href="http://faculty.evansville.edu/ck6/integer/intersp.html">Interspersions and Dispersions</a>.
%H Clark Kimberling, <a href="http://dx.doi.org/10.1090/S0002-9939-1993-1111434-0">Interspersions and dispersions</a>, Proceedings of the American Mathematical Society, Vol. 117, No. 2 (1993), pp. 313-321.
%H N. J. A. Sloane, <a href="/classic.html#WYTH">Classic Sequences</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4,1).
%F a(n) = A001906(n) - 1.
%F G.f.: x*(2 - x)/((1 - x)*(1 - 3*x + x^2)). a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3). - _R. J. Mathar_, Dec 15 2008; adapted to the offset by _Bruno Berselli_, Apr 19 2011
%F a(n) = Fibonacci(4*n+2) mod Fibonacci(2*n+2). - _Gary Detlefs_, Nov 22 2010
%F a(n+1) = Sum_{k=0..n} Fibonacci(2*k+3). - _Gary Detlefs_, Dec 24 2010
%F a(n) = Sum_{i=1..n} A112844(i). - _R. J. Mathar_, Apr 19 2011
%F a(n) = floor(Fibonacci(2*n+2) - Fibonacci(n+1)^2/Fibonacci(2*n+2)). - _Gary Detlefs_, Dec 21 2012
%F From _Peter Bala_, Nov 14 2021: (Start)
%F a(n) = Fibonacci(2*n+4)*(Fibonacci(2*n+1) - 1)/(Fibonacci(2*n+3) - 1).
%F a(n)= -2 + Sum_{k = 1..2*n+3} (-1)^(k+1)*Fibonacci(k). (End)
%p g:=z/(1-3*z+z^2): gser:=series(g, z=0, 43): seq(abs(coeff(gser, z, n)-1), n=1..26); # _Zerinvary Lajos_, Mar 22 2009
%p with(combinat):seq(fibonacci(4*n+2) mod fibonacci(2*n+2),n=0..25);
%t Fibonacci[2*Range[0, 5!]] - 1 (* _Vladimir Joseph Stephan Orlovsky_, May 18 2010 *)
%o (MuPAD) numlib::fibonacci(2*n)-1 $ n = 1..38; // _Zerinvary Lajos_, May 08 2008
%o (Sage) [lucas_number1(n, 3, 1)-1 for n in range(1, 27)] # _Zerinvary Lajos_, Dec 07 2009
%o (Magma) [Fibonacci(2*n+2)-1: n in [0..30]]; // _Vincenzo Librandi_, Apr 18 2011
%o (Maxima) makelist(fib(2*n+2)-1,n,0,30); /* _Martin Ettl_, Oct 21 2012 */
%Y With different offset: 2nd row of Inverse Stolarsky array A035507.
%Y Cf. A001906, A104326, A112844, A152891 (partial sums).
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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