%I #37 Nov 26 2023 14:31:53

%S 2,5,10,17,29,51,90,158,277,486,853,1497,2627,4610,8090,14197,24914,

%T 43721,76725,134643,236282,414646,727653,1276942,2240877,3932465,

%U 6900995,12110402,21252274,37295141,65448410,114853953,201554637,353703731,620706778

%N Expansion of ( 2+x+2*x^2 ) / ( 1-2*x+x^2-x^3 ).

%C Previous name was: A coin is tossed n times and the resultant strings of H's and T's are arranged in a circular (cyclic) manner (i.e. the outcome of the n-th toss is chained to the outcome of the first toss). Then the above sequence represents the number of strings, out of total possible strings of n tosses (n>1), having no isolated H, (by an isolated H, we mean single 'H' which is preceded and succeeded by a 'T'), when the resultant strings are arranged and studied in circular manner. Illustration: In the following string of 10 tosses, 'HHTHTHTTTH', there are only 2 isolated H's, namely the H's at toss number 4 and 6. whereas in the string 'THTHTHTTTH', there will be 4 isolated H's, namely at toss number 2,4,6 and 10. In the string 'HHTTHHHTTH' there is no isolated H, as the H at the 10th toss when chained to the first toss, will no longer be the isolated H, but a triple H.

%H W. Just and G. A. Enciso, <a href="http://www.ohio.edu/people/just/PAPERS/OrderCoop25.pdf">Ordered Dynamics in Biased and Cooperative Boolean Networks</a>, 2013.

%H 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 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1).

%F If a(k) denotes the k-th term( k>4), of the above sequence then a(k)=2a(k-1)-a(k-2)+a(k-3), with a(2)=2, a(3)=5, a(4)=10. Also the k-th term, a(k)( k>5), of this sequence, can be obtained by the formula, a(k)=a(k-1)+a(k-2)+a(k-4), (previous 4 terms are needed), where a(2)=2, a(3)=5, a(4)=10, a(5)=17.

%F a(n) = P(2*n + 4) where P is the Perrin sequence (A001608). a(n) is asymptotic to r^(n+2) where r is the real root of x^3 -2*x^2 +x -1 (A109134). For n>2, a(n) = round(r^(n+2)). - _Gerald McGarvey_, Jan 12 2008

%F G.f.: ( -2-x-2*x^2 ) / ( -1+2*x-x^2+x^3 ). - _R. J. Mathar_, Aug 10 2012

%t CoefficientList[ Series[(-2 - x - 2*x^2)/(-1 + 2*x - x^2 + x^3), {x, 0, 34}], x] (* _Robert G. Wilson v_, Jul 10 2013 *)

%t LinearRecurrence[{2, -1, 1}, {2, 5, 10}, 35] (* _Robert G. Wilson v_, Jul 10 2013 *)

%t Table[RootSum[-1 + # - 2 #^2 + #^3 &, #^(n + 2) &], {n, 0, 20}] (* _Eric W. Weisstein_, Nov 26 2023 *)

%K nonn,easy

%O 0,1

%A Mrs. J. P. Shiwalkar (jyotishiwalkar(AT)rediffmail.com) and Mr. M. N. Deshpande (dpratap_ngp(AT)sancharnet.in), Aug 25 2005

%E Shorter name from _Joerg Arndt_, Sep 03 2013