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 A061667 #50 Dec 25 2024 21:01:00
%S A061667 1,3,9,26,73,201,546,1469,3925,10434,27633,72977,192322,506037,
%T A061667 1329885,3491810,9161929,24026745,62983842,165055853,432445861,
%U A061667 1132806018,2967020769,7770353441,20348233858,53282736741,139516753581,365301078434,956453590585
%N A061667 a(n) = Fibonacci(2*n+1) - 2^(n-1).
%C A061667 Number of cells in the bottom row of all directed column-convex polyominoes of area n+1.
%C A061667 Also the binomial transform of A000071 (after removing its 2 leading zeros). - _R. J. Mathar_, Nov 04 2008
%C A061667 Equals row sums of triangle A147293. - _Gary W. Adamson_, Nov 05 2008
%H A061667 Harry J. Smith, <a href="/A061667/b061667.txt">Table of n, a(n) for n = 1..200</a>
%H A061667 Elena Barcucci, Renzo Pinzani, and Renzo Sprugnoli, <a href="http://dx.doi.org/10.1007/3-540-56610-4_71">Directed column-convex polyominoes by recurrence relations</a>, Lecture Notes in Computer Science, No. 668, Springer, Berlin (1993), pp. 282-298.
%H A061667 Alexander Burstein and Toufik Mansour, <a href="https://arxiv.org/abs/math/0112281">Words restricted by 3-letter generalized multipermutation patterns</a>, arXiv:math/0112281 [math.CO], 2001.
%H A061667 Alexander Burstein and Toufik Mansour, <a href="http://dx.doi.org/10.1007/s000260300000">Words restricted by 3-letter generalized multipermutation patterns</a>, Annals. Combin., 7 (2003), 1-14; see Th. 3.8.
%H A061667 Manosij Ghosh Dastidar and Michael Wallner, <a href="https://arxiv.org/abs/2402.17849">Bijections and congruences involving lattice paths and integer compositions</a>, arXiv:2402.17849 [math.CO], 2024. See p. 22.
%H A061667 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-7,2).
%F A061667 G.f.: x*(1-x)^2/((1-2*x)*(1-3*x+x^2)). - corrected by _Philip B. Zhang_, Nov 28 2014
%F A061667 a(n) = Sum_{k=0..n+1} C(n+1, k)*sum{j=0..floor(k/2), Fibonacci(k-2j)}. - _Paul Barry_, Apr 17 2005
%F A061667 a(n) = 2*A001906(n+1)-A001906(n)-A000079(n). - _R. J. Mathar_, Nov 16 2007
%F A061667 From _Colin Barker_, Jun 05 2017: (Start)
%F A061667 a(n) = 2^(-1-n)*(-5*4^n - (3-sqrt(5))^n*(-5+sqrt(5)) + (3+sqrt(5))^n*(5+sqrt(5))) / 5.
%F A061667 a(n) = 5*a(n-1) - 7*a(n-2) + 2*a(n-3) for n>3. (End)
%t A061667 Table[Fibonacci[2 n + 1] - 2^(n - 1), {n, 1, 100}] (* _Vladimir Joseph Stephan Orlovsky_, Jul 01 2011 *)
%o A061667 (PARI) a(n) = { fibonacci(2*n + 1) - 2^(n - 1) } \\ _Harry J. Smith_, Jul 26 2009
%o A061667 (PARI) Vec(x*(1-x)^2/((1-2*x)*(1-3*x+x^2)) + O(x^50)) \\ _Michel Marcus_, Nov 29 2014
%Y A061667 Cf. A000045.
%Y A061667 Cf. A147293. - _Gary W. Adamson_, Nov 05 2008
%K A061667 nonn,easy
%O A061667 1,2
%A A061667 _Emeric Deutsch_, Jun 16 2001
%E A061667 Offset changed from 0 to 1 by _Harry J. Smith_, Jul 26 2009