A266977 Number of ON (black) cells in the n-th iteration of the "Rule 78" elementary cellular automaton starting with a single ON (black) cell.

1, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36

Offset

0,2

Comments

Also, a(n-1) is the number of topologically inequivalent opening moves in the Sprouts game on n nodes [Browne]. - Andrey Zabolotskiy, Feb 12 2020

Also the number of symmetrically distinct faces in the 1 x 1 x (n+1) polycube. - Eric W. Weisstein, Sep 02 2022

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

Cameron B. Browne, Boundary Matching for Interactive Sprouts, in: ACG 2015, pp. 147-159, LNCS 9525, Springer, doi:10.1007/978-3-319-27992-3_14.

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 entries for linear recurrences with constant coefficients, signature (1,1,-1).

Index to Elementary Cellular Automata

Formula

From Colin Barker, Jan 08 2016 and Apr 16 2019: (Start)

a(n) = (2*n+(-1)^n+7)/4 for n>0.

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

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

From Stefano Spezia, Aug 08 2021: (Start)

E.g.f.: ((4 + x)*cosh(x) + (3 + x)*sinh(x))/2 - 1.

a(n) = 2 + A004526(n) for n > 0. (End)

Mathematica

rule=78; 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 *) Table[Total[catri[[k]]], {k, 1, rows}] (* Number of Black cells in stage n *)

Crossrefs

Cf. A004526, A266974.

Sequence in context: A005410 A120835 A091374 * A370457 A361234 A065603

Adjacent sequences: A266974 A266975 A266976 * A266978 A266979 A266980

Keyword

nonn,easy

Author

Robert Price, Jan 07 2016

Status

approved