A266324 Decimal representation of the n-th iteration of the "Rule 19" elementary cellular automaton starting with a single ON (black) cell.

1, 5, 0, 127, 0, 2047, 0, 32767, 0, 524287, 0, 8388607, 0, 134217727, 0, 2147483647, 0, 34359738367, 0, 549755813887, 0, 8796093022207, 0, 140737488355327, 0, 2251799813685247, 0, 36028797018963967, 0, 576460752303423487, 0, 9223372036854775807, 0

Offset

0,2

References

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Links

Robert Price, Table of n, a(n) for n = 0..500

Eric Weisstein's World of Mathematics, Elementary Cellular Automaton

Index entries for sequences related to cellular automata

Index to Elementary Cellular Automata

Index entries for linear recurrences with constant coefficients, signature (0,17,0,-16).

Formula

From Colin Barker, Dec 28 2015 and Apr 15 2019: (Start)

a(n) = 17*a(n-2) - 16*a(n-4) for n>5.

G.f.: (1+5*x-17*x^2+42*x^3+16*x^4-32*x^5) / ((1-x)*(1+x)*(1-4*x)*(1+4*x)). (End)

a(n) = (1-(-1)^n)*(4*16^floor(n/2)-1/2) for n>1. - Bruno Berselli, Dec 29 2015

a(n) = (2*4^n - 1)*(n mod 2) + 0^n - 2*0^abs(n-1). - Karl V. Keller, Jr., Sep 02 2021

E.g.f.: 1 - 2*x - sinh(x) + 2*sinh(4*x). - Stefano Spezia, Sep 03 2021

Mathematica

rule=19; rows=20; ca=CellularAutomaton[rule, {{1}, 0}, rows-1, {All, All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]], {rows-k+1, rows+k-1}], {k, 1, rows}]; (* Truncated list of each row *) Table[FromDigits[catri[[k]], 2], {k, 1, rows}] (* Decimal Representation of Rows *)

Prog

(Magma) [n le 1 select 5^n else (1-(-1)^n)*(4*16^Floor(n/2)-1/2): n in [0..40]]; // Bruno Berselli, Dec 29 2015
(Python) print([(2*4**n - 1)*(n%2) + 0**n - 2*0**abs(n-1) for n in range(50)]) # Karl V. Keller, Jr., Sep 02 2021

Crossrefs

Cf. A266155, A266323.

Sequence in context: A122045 A294314 A347599 * A073911 A157302 A275759

Adjacent sequences: A266321 A266322 A266323 * A266325 A266326 A266327

Keyword

nonn,easy

Author

Robert Price, Dec 27 2015

Status

approved