A334244 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.

1, 15, 81, 959, 5185, 61391, 331857, 3929023, 21238849, 251457487, 1359286353, 16093279167, 86994326593, 1029969866703, 5567636901969, 65918071468991, 356328761726017, 4218756574015439, 22805040750465105, 270000420736988095, 1459522608029766721, 17280026927167238095

Offset: 1

Author

Pietro Tiaraju Giavarina dos Santos, Apr 20 2020

Keywords

• nonn

Comments

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

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (0,64,0,1,0,-64).

Crossrefs

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

Programs

Mathematica

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 *)

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

Formula

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).

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

From G. C. Greubel, May 29 2020: (Start)

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

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

(End)

a(n) = 64*a(n-2) + a(n-4) - 64*a(n-6) for n>6. - Colin Barker, Jun 10 2020