login
A266723
Total number of ON (black) cells after n iterations of the "Rule 59" elementary cellular automaton starting with a single ON (black) cell.
3
1, 3, 5, 11, 13, 23, 25, 39, 41, 59, 61, 83, 85, 111, 113, 143, 145, 179, 181, 219, 221, 263, 265, 311, 313, 363, 365, 419, 421, 479, 481, 543, 545, 611, 613, 683, 685, 759, 761, 839, 841, 923, 925, 1011, 1013, 1103, 1105, 1199, 1201, 1299, 1301, 1403, 1405
OFFSET
0,2
REFERENCES
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
FORMULA
Conjectures from Colin Barker, Jan 05 2016 and Apr 18 2019: (Start)
a(n) = (3+(-1)^n-2*(-3+(-1)^n)*n+2*n^2)/4.
a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5) for n>4.
G.f.: (1+2*x-x^2)*(1+x^2) / ((1-x)^3*(1+x)^2).
(End)
MATHEMATICA
rule=59; 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 *)
CROSSREFS
Cf. A266716.
Sequence in context: A295243 A179017 A078971 * A129096 A079448 A045407
KEYWORD
nonn
AUTHOR
Robert Price, Jan 03 2016
STATUS
approved