This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A001588 M2279 N0901 #92 Jun 13 2025 18:26:52
%S A001588 1,3,3,5,7,11,17,27,43,69,111,179,289,467,755,1221,1975,3195,5169,
%T A001588 8363,13531,21893,35423,57315,92737,150051,242787,392837,635623,
%U A001588 1028459,1664081,2692539,4356619,7049157,11405775,18454931,29860705,48315635
%N A001588 a(n) = a(n-1) + a(n-2) - 1.
%D A001588 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001588 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001588 T. D. Noe, <a href="/A001588/b001588.txt">Table of n, a(n) for n = 0..500</a>
%H A001588 Massimiliano Fasi and Gian Maria Negri Porzio, <a href="http://eprints.maths.manchester.ac.uk/id/eprint/2768">Determinants of Normalized Bohemian Upper Hessemberg Matrices</a>, University of Manchester (England, 2019).
%H A001588 Martin Griffiths, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Griffiths/gr48.html">On a Matrix Arising from a Family of Iterated Self-Compositions</a>, Journal of Integer Sequences, 18 (2015), #15.11.8.
%H A001588 J. A. H. Hunter and F. D. Parker, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/5-3/elementary5-3.pdf">Problem B-100</a>, Fib. Quart., 5 (1967), p. 288.
%H A001588 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A001588 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A001588 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1).
%F A001588 From _Henry Bottomley_, Feb 20 2001: (Start)
%F A001588 a(n) = 2*Fibonacci(n) + 1 = A000045(n) + A001611(n).
%F A001588 G.f.: (1+x-3x^2)/(1-2*x+x^3). (End)
%F A001588 If n>=4, a(n) = floor(Phi*a(n-1)); Phi = (1 + sqrt(5))/2. - _Philippe Deléham_, Aug 08 2003
%F A001588 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
%p A001588 A001588:=-(-1-z+3*z**2)/(z-1)/(z**2+z-1); # conjectured by _Simon Plouffe_ in his 1992 dissertation
%p A001588 with(combinat): seq(fibonacci(n-2) + fibonacci(n+1) + 1, n = 0..35); # _Zerinvary Lajos_, Feb 01 2008
%t A001588 Fibonacci[Range[0,100]]*2+1 (* _Vladimir Joseph Stephan Orlovsky_, Mar 19 2010 *)
%t A001588 nxt[{a_,b_}]:={b,a+b-1}; NestList[nxt,{1,3},40][[;;,1]] (* _Harvey P. Dale_, Jun 13 2025 *)
%o A001588 (PARI) a(n)=2*fibonacci(n)+1 \\ _Charles R Greathouse IV_, Apr 06 2016
%Y A001588 Cf. A000045, A001611, A001906.
%K A001588 nonn,easy
%O A001588 0,2
%A A001588 _N. J. A. Sloane_