A266302 Decimal representation of the n-th iteration of the "Rule 15" elementary cellular automaton starting with a single ON (black) cell.

1, 6, 1, 126, 1, 2046, 1, 32766, 1, 524286, 1, 8388606, 1, 134217726, 1, 2147483646, 1, 34359738366, 1, 549755813886, 1, 8796093022206, 1, 140737488355326, 1, 2251799813685246, 1, 36028797018963966, 1, 576460752303423486, 1, 9223372036854775806, 1

Offset

0,2

References

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

Links

Robert Price, Table of n, a(n) for n = 0..999

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) = (3*(-1)^n+2^(2*n+1)-(-1)^n*2^(2*n+1)-1)/2.

a(n) = 17*a(n-2)-16*a(n-4) for n>3.

G.f.: (1+6*x-16*x^2+24*x^3) / ((1-x)*(1+x)*(1-4*x)*(1+4*x)).

(End)

a(n) = 2*4^n - 2 for odd n; a(n) = 1 for even n. - Karl V. Keller, Jr., Aug 31 2021

E.g.f.: cosh(x) - 2*sinh(x) + 2*sinh(4*x). - Stefano Spezia, Sep 01 2021

Mathematica

rule=15; 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

(Python) print([2*4**n - 2 if n%2 else 1 for n in range(50)]) # Karl V. Keller, Jr., Aug 31 2021

Crossrefs

Cf. A266300, A266301.

Sequence in context: A331557 A352058 A303675 * A352012 A183284 A224476

Adjacent sequences: A266299 A266300 A266301 * A266303 A266304 A266305

Keyword

nonn,easy

Author

Robert Price, Dec 26 2015

Status

approved