A263807 Total number of ON (black) cells after n iterations of the "Rule 157" elementary cellular automaton starting with a single ON (black) cell.

1, 3, 6, 11, 17, 25, 34, 45, 57, 71, 86, 103, 121, 141, 162, 185, 209, 235, 262, 291, 321, 353, 386, 421, 457, 495, 534, 575, 617, 661, 706, 753, 801, 851, 902, 955, 1009, 1065, 1122, 1181, 1241, 1303, 1366, 1431, 1497, 1565, 1634, 1705, 1777, 1851, 1926

Offset: 0

Author

Robert Price, Jan 17 2016

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..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. A263804.

Programs

Mathematica

rule=157; 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 *)

Formula

Conjectures from Colin Barker, Jan 20 2016 and Apr 16 2019: (Start)

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

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

(End)