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 A007039 M0241 #61 Mar 09 2025 12:25:30
%S A007039 2,2,2,6,12,20,30,46,74,122,200,324,522,842,1362,2206,3572,5780,9350,
%T A007039 15126,24474,39602,64080,103684,167762,271442,439202,710646,1149852,
%U A007039 1860500,3010350,4870846,7881194,12752042,20633240,33385284,54018522,87403802,141422322
%N A007039 Number of cyclic binary n-bit strings with no alternating substring of length > 2.
%C A007039 _John W. Layman_ observes that the second differences give the sequence shifted to the right.
%D A007039 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007039 Vincenzo Librandi, <a href="/A007039/b007039.txt">Table of n, a(n) for n = 1000</a>
%H A007039 Z. Agur, A. S. Fraenkel, and S. T. Klein, <a href="http://dx.doi.org/10.1016/0012-365X(88)90005-2">The number of fixed points of the majority rule</a>, Discr. Math., 70 (1988), 295-302.
%H A007039 Matthew Macauley, Jon McCammond, and Henning S. Mortveit, <a href="http://www.emis.de/journals/JACO/Volume33_1/hgv665924j44t770.html">Dynamics groups of asynchronous cellular automata</a>, Journal of Algebraic Combinatorics, Vol 33, No 1 (2011), pp. 11-35.
%H A007039 A. McLeod and W. O. J. Moser, <a href="http://www.jstor.org/stable/27642988">Counting cyclic binary strings</a>, Math. Mag., 80 (No. 1, 2007), 29-37.
%H A007039 W. O. J. Moser, <a href="/A007040/a007040.pdf">On cyclic binary n-bit strings</a>, Report from the Department of Mathematics and Statistics, McGill University, 1991. (Annotated scanned copy)
%H A007039 W. O. J. Moser, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/31-1/moser.pdf">Cyclic binary strings without long runs of like (alternating) bits</a>, Fibonacci Quart. 31 (1993), no. 1, 2-6.
%H A007039 E. Munarini and N. Z. Salvi, <a href="http://www.emis.de/journals/INTEGERS/papers/d19/d19.Abstract.html">Circular Binary Strings without Zigzags</a>, Integers: Electronic Journal of Combinatorial Number Theory 3 (2003), #A19. See relation (8) on page 5.
%H A007039 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,1).
%F A007039 For n >= 5, a(n) = 2a(n-1) - a(n-2) + a(n-4). - _David W. Wilson_
%F A007039 G.f.: 2*x*(1+x)*(1-2*x+2*x^2)/((1-x+x^2)*(1-x-x^2)). - _Colin Barker_, Mar 28 2012
%F A007039 a(n) = A000032(n) + A057079(n + 1). - _John M. Campbell_, Dec 29 2016
%F A007039 a(n) = abs(A111734(n)). - _Alois P. Heinz_, Oct 08 2017
%F A007039 E.g.f.: 2*exp(x/2)*(cos(sqrt(3)*x/2) + cosh(sqrt(5)*x/2)) - 4. - _Stefano Spezia_, Mar 09 2025
%e A007039 G.f. = 2*x + 2*x^2 + 2*x^3 + 6*x^4 + 12*x^5 + 20*x^6 + 30*x^7 + 46*x^8 + ...
%t A007039 CoefficientList[Series[2*(1+x)*(1-2*x+2*x^2)/((1-x+x^2)*(1-x-x^2)),{x,0,40}],x] (* _Vincenzo Librandi_, Apr 16 2012 *)
%t A007039 a[n_ /; n<4] = 2; a[4] = 6; a[n_] := a[n] = 2*a[n-1] - a[n-2] + a[n-4]; Array[a, 39] (* _Jean-François Alcover_, Oct 08 2017 *)
%o A007039 (PARI) Vec(2*x*(1-x+2*x^3)/((1-x-x^2)*(1-x+x^2))+O(x^66)) \\ _Joerg Arndt_, Oct 27 2015
%Y A007039 Cf. A000032, A007040, A057079, A111734.
%K A007039 nonn,easy,eigen
%O A007039 1,1
%A A007039 _N. J. A. Sloane_