A267049 Total number of OFF (white) cells after n iterations of the "Rule 91" elementary cellular automaton starting with a single ON (black) cell.

0, 1, 4, 7, 11, 14, 22, 25, 37, 40, 56, 59, 79, 82, 106, 109, 137, 140, 172, 175, 211, 214, 254, 257, 301, 304, 352, 355, 407, 410, 466, 469, 529, 532, 596, 599, 667, 670, 742, 745, 821, 824, 904, 907, 991, 994, 1082, 1085, 1177, 1180, 1276, 1279, 1379, 1382

Offset: 0

Author

Robert Price, Jan 09 2016

Keywords

• nonn

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

Crossrefs

Cf. A267015.

Programs

Mathematica

rule=91; 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 *) nwc=Table[Length[catri[[k]]]-nbc[[k]],{k,1,rows}]; (* Number of White cells in stage n *) Table[Total[Take[nwc,k]],{k,1,rows}] (* Number of White cells through stage n *)

Formula

Conjectures from Colin Barker, Jan 11 2016 and Apr 19 2019: (Start)

a(n) = (n^2+(-1)^n*(n-3)+5)/2 for n>1.

a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5) for n>6.

G.f.: x*(1+3*x+x^2-2*x^3-2*x^4+3*x^5) / ((1-x)^3*(1+x)^2).

(End)