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A109377 Expansion of ( 2+x+2*x^2 ) / ( 1-2*x+x^2-x^3 ). 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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

Table of n, a(n) for n=0..34.

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).

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

MATHEMATICA

CoefficientList[ Series[(-2 - x - 2*x^2)/(-1 + 2*x - x^2 + x^3), {x, 0, 34}], x] (* or *) LinearRecurrence[{2, -1, 1}, {2, 5, 10}, 35] (* Robert G. Wilson v, Jul 10 2013 *)

CROSSREFS

Sequence in context: A071602 A046485 A294562 * A109472 A172167 A173060

Adjacent sequences: A109374 A109375 A109376 * A109378 A109379 A109380

KEYWORD

nonn,easy

AUTHOR

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

EXTENSIONS

Shorter name from Joerg Arndt, Sep 03 2013

STATUS

approved

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Last modified December 9 19:36 EST 2022. Contains 358703 sequences. (Running on oeis4.)