A035508 a(n) = Fibonacci(2*n+2) - 1.

0, 2, 7, 20, 54, 143, 376, 986, 2583, 6764, 17710, 46367, 121392, 317810, 832039, 2178308, 5702886, 14930351, 39088168, 102334154, 267914295, 701408732, 1836311902, 4807526975, 12586269024, 32951280098, 86267571271, 225851433716

Offset

0,2

Comments

Except for 0, numbers whose dual Zeckendorf representation (A104326) has the same number of 0's as 1's. - Amiram Eldar, Mar 22 2021

Links

Table of n, a(n) for n=0..27.

Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]

Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.

Clark Kimberling, Interspersions and Dispersions.

Clark Kimberling, Interspersions and dispersions, Proceedings of the American Mathematical Society, Vol. 117, No. 2 (1993), pp. 313-321.

N. J. A. Sloane, Classic Sequences.

Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

Formula

a(n) = A001906(n) - 1.

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

a(n) = Fibonacci(4*n+2) mod Fibonacci(2*n+2). - Gary Detlefs, Nov 22 2010

a(n+1) = Sum_{k=0..n} Fibonacci(2*k+3). - Gary Detlefs, Dec 24 2010

a(n) = Sum_{i=1..n} A112844(i). - R. J. Mathar, Apr 19 2011

a(n) = floor(Fibonacci(2*n+2) - Fibonacci(n+1)^2/Fibonacci(2*n+2)). - Gary Detlefs, Dec 21 2012

From Peter Bala, Nov 14 2021: (Start)

a(n) = Fibonacci(2*n+4)*(Fibonacci(2*n+1) - 1)/(Fibonacci(2*n+3) - 1).

a(n)= -2 + Sum_{k = 1..2*n+3} (-1)^(k+1)*Fibonacci(k). (End)

Maple

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
with(combinat):seq(fibonacci(4*n+2) mod fibonacci(2*n+2), n=0..25);

Mathematica

Fibonacci[2*Range[0, 5!]] - 1 (* Vladimir Joseph Stephan Orlovsky, May 18 2010 *)

Prog

(MuPAD) numlib::fibonacci(2*n)-1 $ n = 1..38; // Zerinvary Lajos, May 08 2008
(Sage) [lucas_number1(n, 3, 1)-1 for n in range(1, 27)] # Zerinvary Lajos, Dec 07 2009
(Magma) [Fibonacci(2*n+2)-1: n in [0..30]]; // Vincenzo Librandi, Apr 18 2011
(Maxima) makelist(fib(2*n+2)-1, n, 0, 30); /* Martin Ettl, Oct 21 2012 */

Crossrefs

With different offset: 2nd row of Inverse Stolarsky array A035507.

Cf. A001906, A104326, A112844, A152891 (partial sums).

Sequence in context: A348571 A128183 A027418 * A018033 A000149 A080041

Adjacent sequences: A035505 A035506 A035507 * A035509 A035510 A035511

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved