A263511 Total number of ON (black) cells after n iterations of the "Rule 155" elementary cellular automaton starting with a single ON (black) cell.

1, 3, 6, 12, 19, 29, 40, 54, 69, 87, 106, 128, 151, 177, 204, 234, 265, 299, 334, 372, 411, 453, 496, 542, 589, 639, 690, 744, 799, 857, 916, 978, 1041, 1107, 1174, 1244, 1315, 1389, 1464, 1542, 1621, 1703, 1786, 1872, 1959, 2049, 2140, 2234, 2329, 2427

Offset: 0

Author

Robert Price, Jan 17 2016

Keywords

• nonn
• easy

Comments

Conjecture: it appears that these number also describe the number of numerically different eigenvalues of the triangular grid graphs with an odd number of edges on each side of the outer triangle, starting with |eigenvalues(T_3)|=6, |eigenvalues(T_5)|=12, etc. (checked up to |eigenvalues(T_13)|=54). - Michael Terhoeven, Jul 12 2025

References

• Stephen 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
• Eric Weisstein's World of Mathematics, Triangular Grid Graph
• Stephen Wolfram, A New Kind of Science
• Index entries for sequences related to cellular automata
• Index to Elementary Cellular Automata
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Crossrefs

Cf. A263243.

Programs

Magma

I:=[1,3,6,12]; [n le 4 select I[n] else 2*Self(n-1)-2*Self(n-3)+Self(n-4): n in [1..60]]; // Vincenzo Librandi, Jan 18 2016

Mathematica

rule=155; 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 *)
LinearRecurrence[{2, 0, -2, 1}, {1, 3, 6, 12}, 50] (* Vincenzo Librandi, Jan 18 2016 *)

Formula

G.f.: (1+x+2*x^3)/((1-x)^3*(1+x)). - Vincenzo Librandi, Jan 18 2016

a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n>3. - Vincenzo Librandi, Jan 18 2016