A263511 Total number of ON (black) cells after n iterations of the "Rule 155" elementary cellular automaton starting with a single ON (black) cell.

1, 3, 6, 12, 19, 29, 40, 54, 69, 87, 106, 128, 151, 177, 204, 234, 265, 299, 334, 372, 411, 453, 496, 542, 589, 639, 690, 744, 799, 857, 916, 978, 1041, 1107, 1174, 1244, 1315, 1389, 1464, 1542, 1621, 1703, 1786, 1872, 1959, 2049, 2140, 2234, 2329, 2427

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..1000

Eric Weisstein's World of Mathematics, Elementary Cellular Automaton

S. Wolfram, A New Kind of Science

Index entries for sequences related to cellular automata

Index to Elementary Cellular Automata

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Formula

G.f.: (1+x+2*x^3)/((1-x)^3*(1+x)). - Vincenzo Librandi, Jan 18 2016

a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n>3. - Vincenzo Librandi, Jan 18 2016

Mathematica

rule=155; 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 *) nbc=Table[Total[catri[[k]]], {k, 1, rows}]; (* Number of Black cells in stage n *) Table[Total[Take[nbc, k]], {k, 1, rows}] (* Number of Black cells through stage n *)
LinearRecurrence[{2, 0, -2, 1}, {1, 3, 6, 12}, 50] (* Vincenzo Librandi, Jan 18 2016 *)

Prog

(Magma) I:=[1, 3, 6, 12]; [n le 4 select I[n] else 2*Self(n-1)-2*Self(n-3)+Self(n-4): n in [1..60]]; // Vincenzo Librandi, Jan 18 2016

Crossrefs

Cf. A263243.

Sequence in context: A318872 A267591 A101423 * A180622 A125851 A351621

Adjacent sequences: A263508 A263509 A263510 * A263512 A263513 A263514

Keyword

nonn,easy

Author

Robert Price, Jan 17 2016

Status

approved