A001588 a(n) = a(n-1) + a(n-2) - 1. (Formerly M2279 N0901)

1, 3, 3, 5, 7, 11, 17, 27, 43, 69, 111, 179, 289, 467, 755, 1221, 1975, 3195, 5169, 8363, 13531, 21893, 35423, 57315, 92737, 150051, 242787, 392837, 635623, 1028459, 1664081, 2692539, 4356619, 7049157, 11405775, 18454931, 29860705, 48315635

Offset

0,2

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

Links

T. D. Noe, Table of n, a(n) for n = 0..500

Massimiliano Fasi, Gian Maria Negri Porzio, Determinants of Normalized Bohemian Upper Hessemberg Matrices, University of Manchester (England, 2019).

Martin Griffiths, On a Matrix Arising from a Family of Iterated Self-Compositions, Journal of Integer Sequences, 18 (2015), #15.11.8.

J. A. H. Hunter and F. D. Parker, Problem B-100, Fib. Quart., 5 (1967), p. 288.

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 (2,0,-1).

Formula

From Henry Bottomley, Feb 20 2001: (Start)

a(n) = 2*Fibonacci(n) + 1 = A000045(n) + A001611(n).

G.f.: (1+x-3x^2)/(1-2*x+x^3). (End)

If n>=4, a(n) = floor(Phi*a(n-1)); Phi = (1 + sqrt(5))/2. - Philippe Deléham, Aug 08 2003

a(n) = F(n-2) + F(n+1) + 1, n >= 0 (where F(n) is the n-th Fibonacci number). - Zerinvary Lajos, Feb 01 2008

a(n) = 1 + (2/5)*((1/2) + (1/2)*sqrt(5))^n*sqrt(5) - (2/5)*sqrt(5)*((1/2) - (1/2)*sqrt(5))^n, with n >= 0. - Paolo P. Lava, Nov 21 2008

Maple

A001588:=-(-1-z+3*z**2)/(z-1)/(z**2+z-1); # conjectured by Simon Plouffe in his 1992 dissertation
with(combinat): seq(fibonacci(n-2) + fibonacci(n+1) + 1, n = 0..35); # Zerinvary Lajos, Feb 01 2008

Mathematica

Fibonacci[Range[0, 100]]*2+1 (* Vladimir Joseph Stephan Orlovsky, Mar 19 2010 *)

Prog

(PARI) a(n)=2*fibonacci(n)+1 \\ Charles R Greathouse IV, Apr 06 2016

Crossrefs

Sequence in context: A323529 A328222 A306276 * A107029 A352912 A240180

Adjacent sequences: A001585 A001586 A001587 * A001589 A001590 A001591

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved