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%I #14 Jul 03 2023 08:41:36
%S 49,97,145,2497,7153,14113,133969,477313,1154737,7585249,30496273,
%T 85923649,450015601,1913836705,6038171857,27638920705,119503082545,
%U 409335331681,1736003525521,7472151487681,27120247408369,110448416633377,469111688042065,1770883563643777
%N The number of different configurations of an n-block of a shift space with k symbols where each symbol but the first must appear isolated and separated from others by an block of length at least m made of first symbol. Here k=49 and m=2.
%C For k=2 is the number of different configurations of an n-bits string where each 1 is isolated and separated by at least m zeros.
%C Limit as n-> infinity (a(n+1)/a(n)) -> 4.
%H R. Tonelli, <a href="https://arxiv.org/abs/0708.4370">Fibonacci-like sequences and shift spaces in symbolic dynamics</a>, arXiv:0708.4370, [math.DS], 2007.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 48).
%F a(n) = a(n-1) + (k-1)*a(n-m-1), where k=49, m=2, n>=4, with a(1) = 49, a(2) = 97, a(3) = 145.
%F G.f.: -x*(49+48*x+48*x^2)/(4*x-1)/(12*x^2+3*x+1). - _R. J. Mathar_, Nov 14 2007
%t Rest[CoefficientList[Series[-x*(49+48*x+48*x^2)/(4*x-1)/(12*x^2+3*x+1), {x, 0, 50}], x]] (* _G. C. Greubel_, May 02 2017 *)
%o (PARI) x='x+O('x^50); Vec(-x*(49+48*x+48*x^2)/(4*x-1)/(12*x^2+3*x+1)) \\ _G. C. Greubel_, May 02 2017
%Y Cf. A000045, A000930, A131600.
%K easy,nonn
%O 1,1
%A R. Tonelli (roberto.tonelli(AT)dsf.unica.it), Aug 31 2007