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Decimal representation of n-th iteration of the one-dimensional cellular automaton defined by Rule 950, based on the 4-celled von Neumann neighborhood starting with a single black cell.
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%I #31 Jul 16 2020 06:05:12

%S 1,15,81,959,5185,61391,331857,3929023,21238849,251457487,1359286353,

%T 16093279167,86994326593,1029969866703,5567636901969,65918071468991,

%U 356328761726017,4218756574015439,22805040750465105,270000420736988095,1459522608029766721,17280026927167238095

%N Decimal representation of n-th iteration of the one-dimensional cellular automaton defined by Rule 950, based on the 4-celled von Neumann neighborhood starting with a single black cell.

%C a(n) is the decimal representation of the n-th step based on a simple initial condition, when a(1) = 1.

%H Colin Barker, <a href="/A334244/b334244.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,64,0,1,0,-64).

%F a(n) = (-1040 + 130*(-1)^n - (504 + 504*i)*(-i)^n - (504 - 504*i)*i^n + 1607*2^(3*n) + 311*(-1)^n*2^(3*n))/8190 where i = sqrt(-1).

%F G.f.: (1 + 15*x + 17*x^2 - x^3)/(1 - 64*x^2 - x^4 + 64*x^6).

%F From _G. C. Greubel_, May 29 2020: (Start)

%F a(n) = ( (1607 + 311*(-1)^n)*8^n - (1040 - 130*(-1)^n) - 1008*sqrt(2)*cos((2*n-1)*Pi/4) )/8190.

%F E.g.f.: (959*cosh(8*x) + 648*sinh(8*x) - 455*cosh(x) - 585*sinh(x) - 504*(cos(x) + sin(x)) )/4095.

%F (End)

%F a(n) = 64*a(n-2) + a(n-4) - 64*a(n-6) for n>6. - _Colin Barker_, Jun 10 2020

%t Table[((1607 +311*(-1)^n)*8^n -1040 +130*(-1)^n -1008*Sqrt[2]*Cos[(2*n-1)*Pi/4] )/8190, {n, 25}] (* _G. C. Greubel_, May 29 2020 *)

%o (PARI) Vec(x*(1 + 15*x + 17*x^2 - x^3) / ((1 - x)*(1 + x)*(1 - 8*x)*(1 + 8*x)*(1 + x^2)) + O(x^20)) \\ _Colin Barker_, Jun 10 2020

%Y Cf. A118171, A118173 (similar examples from elementary cellular automata).

%K nonn

%O 1,2

%A _Pietro Tiaraju Giavarina dos Santos_, Apr 20 2020