A266223 Total number of OFF (white) cells after n iterations of the "Rule 7" elementary cellular automaton starting with a single ON (black) cell.

0, 1, 6, 6, 15, 15, 28, 28, 45, 45, 66, 66, 91, 91, 120, 120, 153, 153, 190, 190, 231, 231, 276, 276, 325, 325, 378, 378, 435, 435, 496, 496, 561, 561, 630, 630, 703, 703, 780, 780, 861, 861, 946, 946, 1035, 1035, 1128, 1128, 1225, 1225, 1326, 1326, 1431

Offset: 0

Author

Robert Price, Dec 24 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..499
• Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
• Index entries for sequences related to cellular automata
• Index to Elementary Cellular Automata

Crossrefs

Cf. A266216.

Programs

Mathematica

rule=7; 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, Dec 26 2015 and Apr 14 2019: (Start)

a(n) = 1/2*(n+1)*(n+(-1)^n+1) for n>0.

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

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

(End)

Extensions

Conjectures from Colin Barker, Apr 14 2019