A109377 Expansion of ( 2+x+2*x^2 ) / ( 1-2*x+x^2-x^3 ).

2, 5, 10, 17, 29, 51, 90, 158, 277, 486, 853, 1497, 2627, 4610, 8090, 14197, 24914, 43721, 76725, 134643, 236282, 414646, 727653, 1276942, 2240877, 3932465, 6900995, 12110402, 21252274, 37295141, 65448410, 114853953, 201554637, 353703731, 620706778

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

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.

Links

• W. Just and G. A. Enciso, Ordered Dynamics in Biased and Cooperative Boolean Networks, 2013.
• Matthew Macauley, Jon McCammond, and Henning S. Mortveit, Dynamics groups of asynchronous cellular automata, Journal of Algebraic Combinatorics, Vol 33, No 1 (2011), pp. 11-35.
• Index entries for linear recurrences with constant coefficients, signature (2,-1,1).

Programs

Mathematica

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 *)
LinearRecurrence[{2, -1, 1}, {2, 5, 10}, 35] (* Robert G. Wilson v, Jul 10 2013 *)
Table[RootSum[-1 + # - 2 #^2 + #^3 &, #^(n + 2) &], {n, 0, 20}] (* Eric W. Weisstein, Nov 26 2023 *)

Formula

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.

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

G.f.: ( -2-x-2*x^2 ) / ( -1+2*x-x^2+x^3 ). - R. J. Mathar, Aug 10 2012

Extensions

Shorter name from Joerg Arndt, Sep 03 2013