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 A007909 #69 Apr 22 2025 12:30:23
%S A007909 1,1,1,3,7,13,25,51,103,205,409,819,1639,3277,6553,13107,26215,52429,
%T A007909 104857,209715,419431,838861,1677721,3355443,6710887,13421773,
%U A007909 26843545,53687091,107374183,214748365,429496729,858993459,1717986919,3435973837,6871947673
%N A007909 Expansion of (1-x)/(1-2*x+x^2-2*x^3).
%C A007909 Equals INVERT transform of (1, 0, 2, 2, 2, ...). - _Gary W. Adamson_, Apr 28 2009
%C A007909 a(n) is the number of compositions (ordered partitions) of n into parts 1 (one kind), and parts >= 3 of three kinds (no parts 2). - _Joerg Arndt_, Apr 22 2025
%D A007909 Kenneth Edwards, Michael A. Allen, A new combinatorial interpretation of the Fibonacci numbers squared, Part II, Fib. Q., 58:2 (2020), 169-177.
%D A007909 M. E. Larsen, Summa Summarum, A. K. Peters, Wellesley, MA, 2007; see p. 38.
%H A007909 M. F. Hasler, <a href="/A007909/b007909.txt">Table of n, a(n) for n = 0..1000</a> (in replacement of a(0..999) indexed 1..1000 from Vincenzo Librandi).
%H A007909 Charles K. Cook and Michael R. Bacon, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AMI_41_from27to39.pdf">Some identities for Jacobsthal and Jacobsthal-Lucas numbers satisfying higher order recurrence relations</a>, Annales Mathematicae et Informaticae, 41 (2013) pp. 27-39.
%H A007909 Shanzhen Gao, Keh-Hsun Chen, <a href="http://worldcomp-proceedings.com/proc/p2014/FCS2696.pdf">Tackling Sequences From Prudent Self-Avoiding Walks</a>, FCS'14, The 2014 International Conference on Foundations of Computer Science.
%H A007909 I. Gessel, <a href="http://www.jstor.org/stable/2974863">Problem 10424</a>, Amer. Math. Monthly, 102 (1995), 70.
%H A007909 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=444">Encyclopedia of Combinatorial Structures 444</a>
%H A007909 Yüksel Soykan, <a href="https://doi.org/10.34198/ejms.5221.297327">Properties of Generalized (r, s, t, u)-Numbers</a>, Earthline J. of Math. Sci. (2021) Vol. 5, No. 2, 297-327.
%H A007909 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,2).
%F A007909 G.f.: (1-x)/(1-2*x+x^2-2*x^3).
%F A007909 a(n) = (1/5)*(2^(n+1)+3*cos(n*Pi/2)+sin(n*Pi/2)).
%F A007909 a(n) = Sum_{k=0..floor((n-1)/3)} binomial(n-k-1, 2*k)*2^k. - _Paul Barry_, Sep 16 2004
%F A007909 a(n) = (1/5)*(2^(n+1) + (-1)^[(n+1)/2] + 2*(-1)^[n/2]). - _Ralf Stephan_, Jun 09 2005
%F A007909 a(n) = 2*a(n-1)-a(n-2)+2*a(n-3). Sequence is identical to its half second differences from the second term; a(n)+a(n+2)=2^(n+1). - _Paul Curtz_, Dec 17 2007
%F A007909 a(n+1) = (2^n)*Sum_{k=1..n} (-1)^floor(k/2)/2^k. - _Yalcin Aktar_, Jul 20 2008
%p A007909 U:=n->(1/5)*(2^(n+1)+3*cos(n*Pi/2)+sin(n*Pi/2)); [seq(U(n),n=0..50)];
%t A007909 CoefficientList[Series[(1-x)/(1-2*x+x^2-2*x^3),{x,0,40}],x] (* _Vincenzo Librandi_, Jun 17 2012 *)
%t A007909 LinearRecurrence[{2,-1,2},{1,1,1},40] (* _Harvey P. Dale_, Jul 26 2016 *)
%o A007909 (Magma) I:=[1, 1, 1]; [n le 3 select I[n] else 2*Self(n-1)-Self(n-2)+2*Self(n-3): n in [1..40]]; // _Vincenzo Librandi_, Jun 17 2012
%o A007909 (PARI) a(n)=2^(n+1)\5+(n%4<2) \\ _M. F. Hasler_, Feb 22 2018
%o A007909 (Python)
%o A007909 def A007909(n): return (2<<n)//5+(n&3<2) # _Chai Wah Wu_, Apr 22 2025
%Y A007909 Cf. A005251, A007679, A007910.
%K A007909 nonn,easy
%O A007909 0,4
%A A007909 Mogens Esrom Larsen (mel(AT)math.ku.dk)
%E A007909 Offset corrected by _M. F. Hasler_, Feb 22 2018