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

Author

Robert Price, Dec 27 2015

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A266155, A266323. Essentially the same as A266218.

Programs

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

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 *)

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

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