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 A069403 #54 Jul 10 2025 04:32:04
%S A069403 1,3,9,25,67,177,465,1219,3193,8361,21891,57313,150049,392835,1028457,
%T A069403 2692537,7049155,18454929,48315633,126491971,331160281,866988873,
%U A069403 2269806339,5942430145,15557484097,40730022147,106632582345,279167724889,730870592323,1913444052081
%N A069403 a(n) = 2*Fibonacci(2*n+1) - 1.
%C A069403 Half the number of n X 3 binary arrays with a path of adjacent 1's and a path of adjacent 0's from top row to bottom row.
%C A069403 Indices of A017245 = 9*n + 7 = 7, 16, 25, 34, for submitted A153819 = 16, 34, 88,. A153819(n) = 9*a(n) + 7 = 18*F(2*n+1) -2; F(n) = Fibonacci = A000045, 2's = A007395. Other recurrence: a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3). - _Paul Curtz_, Jan 02 2009
%H A069403 Colin Barker, <a href="/A069403/b069403.txt">Table of n, a(n) for n = 0..1000</a>
%H A069403 Yurii S. Bystryk, Vitalii L. Denysenko, and Volodymyr I. Ostryk, <a href="http://dx.doi.org/10.13140/RG.2.2.16511.73129">Lune and Lens Sequences</a>, ResearchGate preprint, 2024. See pp. 41, 56.
%H A069403 J. Hietarinta and C.-M. Viallet, <a href="http://dx.doi.org/10.1103/PhysRevLett.81.325">Singularity confinement and chaos in discrete systems</a>, Physical Review Letters 81 (1998), pp. 326-328.
%H A069403 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4,1).
%F A069403 a(0) = 1, a(1) = 3, a(2) = 9, a(3) = 25; a(n) = 3*a(n-1) - 3*a(n-3) + a(n-4).
%F A069403 a(n) = 3*a(n-1) - a(n-2) + 1 for n>1, a(1) = 3, a(0) = 0. - _Reinhard Zumkeller_, May 02 2006
%F A069403 From _R. J. Mathar_, Feb 23 2009: (Start)
%F A069403 a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3).
%F A069403 G.f.: (1-x+x^2)/((1-x)*(1-3*x+x^2)). (End)
%F A069403 a(n) = 1 + 2*Sum_{k=0..n} Fibonacci(2*k) = 1+2*A027941(n). - _Gary Detlefs_, Dec 07 2010
%F A069403 a(n) = (2^(-n)*(-5*2^n -(3-sqrt(5))^n*(-5+sqrt(5)) +(3+sqrt(5))^n*(5+sqrt(5))))/5. - _Colin Barker_, Nov 02 2016
%t A069403 a[n_]:= a[n] = 3a[n-1] - 3a[n-3] + a[n-4]; a[0] = 1; a[1] = 3; a[2] = 9; a[3] = 25; Table[ a[n], {n, 0, 30}]
%t A069403 Table[2*Fibonacci[2*n+1]-1, {n,0,30}] (* _G. C. Greubel_, Apr 22 2018 *)
%t A069403 LinearRecurrence[{4,-4,1},{1,3,9},30] (* _Harvey P. Dale_, Sep 22 2020 *)
%o A069403 (Magma) [2*Fibonacci(2*n+1)-1: n in [0..30]]; // _Vincenzo Librandi_, Apr 18 2011
%o A069403 (PARI) a(n) = 2*fibonacci(2*n+1)-1 \\ _Charles R Greathouse IV_, Jun 11 2015
%o A069403 (PARI) Vec((1-x+x^2)/((1-x)*(1-3*x+x^2)) + O(x^30)) \\ _Colin Barker_, Nov 02 2016
%o A069403 (Sage) [2*fibonacci(2*n+1)-1 for n in (0..30)] # _G. C. Greubel_, Jul 11 2019
%o A069403 (GAP) List([0..30], n-> 2*Fibonacci(2*n+1)-1); # _G. C. Greubel_, Jul 11 2019
%Y A069403 Cf. A000045, A084707.
%Y A069403 Cf. 1 X n A000225, 2 X n A016269, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.
%Y A069403 Equals A052995 - 1.
%Y A069403 Bisection of A001595, A062114, A066983.
%K A069403 nonn,easy
%O A069403 0,2
%A A069403 _R. H. Hardin_, Mar 22 2002
%E A069403 Simpler definition from _Vladeta Jovovic_, Mar 19 2003