A266221 Total number of ON (black) cells after n iterations of the "Rule 7" elementary cellular automaton starting with a single ON (black) cell.

1, 3, 3, 10, 10, 21, 21, 36, 36, 55, 55, 78, 78, 105, 105, 136, 136, 171, 171, 210, 210, 253, 253, 300, 300, 351, 351, 406, 406, 465, 465, 528, 528, 595, 595, 666, 666, 741, 741, 820, 820, 903, 903, 990, 990, 1081, 1081, 1176, 1176, 1275, 1275, 1378, 1378

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..499

Eric Weisstein's World of Mathematics, Elementary Cellular Automaton

Index entries for sequences related to cellular automata

Index to Elementary Cellular Automata

Formula

Conjectures from Colin Barker, Dec 25 2015 and Apr 13 2019: (Start)

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

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

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

(End)

a(n) = A000217(A052928(n+1)) for n>0. - Michel Marcus, Sep 25 2016

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 *) Table[Total[Take[nbc, k]], {k, 1, rows}] (* Number of Black cells through stage n *)

Crossrefs

Cf. A266216.

Sequence in context: A330632 A278832 A168376 * A073709 A085288 A124630

Adjacent sequences: A266218 A266219 A266220 * A266222 A266223 A266224

Keyword

nonn,easy

Author

Robert Price, Dec 24 2015

Extensions

Conjectures from Colin Barker, Apr 13 2019

Status

approved