A001588 a(n) = a(n-1) + a(n-2) - 1.
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
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 and 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).
Programs
-
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 *) nxt[{a_,b_}]:={b,a+b-1}; NestList[nxt,{1,3},40][[;;,1]] (* Harvey P. Dale, Jun 13 2025 *)
-
PARI
a(n)=2*fibonacci(n)+1 \\ Charles R Greathouse IV, Apr 06 2016
Formula
From Henry Bottomley, Feb 20 2001: (Start)
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