A271061 -id:A271061 - cp's OEIS frontend

A211012 Total area of all squares and rectangles after 2^n stages in the toothpick structure of A139250, assuming the toothpicks have length 2.

0, 0, 8, 48, 224, 960, 3968, 16128, 65024, 261120, 1046528, 4190208, 16769024, 67092480, 268402688, 1073676288, 4294836224, 17179607040, 68718952448, 274876858368, 1099509530624, 4398042316800, 17592177655808, 70368727400448, 281474943156224

Offset: 0

Omar E. Pol, Sep 21 2012

All internal regions in the toothpick structure are squares and rectangles. The area of every internal region is a power of 2.
Similar to A271061. - Robert Price, Mar 30 2016
For n=3,5,..., also the number of minimum vertex colorings in the n-sunlet graph. - Eric W. Weisstein, Mar 03 2024

			For n = 3 the area of all squares and rectangles in the toothpick structure after 2^3 stages equals the area of a rectangle of size 8X6, so a(3) = 8*6 = 48.

Colin Barker, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Eric Weisstein's World of Mathematics, Minimum Vertex Coloring.
Eric Weisstein's World of Mathematics, Sunlet Graph.
Index entries for linear recurrences with constant coefficients, signature (6,-8).

Row sums of triangle A211017, n>=1.

Cf. A000079, A006516, A139250, A159786, A160124, A211008, A211016, A211018.

concat(vector(2), Vec(8*x^2/((1-2*x)*(1-4*x)) + O(x^50))) \\ Colin Barker, Mar 30 2016

a(n) = 2^n * (2^n-2) = A000079(n)*(A000079(n) - 2) = A159786(2^n) = 8*A006516(n-1), n>=1.
From Colin Barker, Mar 30 2016: (Start)
G.f.: 8*x^2 / ((1-2*x)*(1-4*x)).
a(n) = 6*a(n-1)-8*a(n-2) for n>2. (End)
E.g.f.: (1 - exp(2*x))^2. - Stefano Spezia, Mar 12 2025

A273335 Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 657", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 4, 48, 224, 960, 3968, 16128, 65024, 261120, 1046528, 4190208, 16769024, 67092480, 268402688, 1073676288, 4294836224

Offset: 0

Robert Price, May 20 2016

Initialized with a single black (ON) cell at stage zero.
Conjecture: Rule 665 also generates this sequence. - Lars Blomberg, Jul 18 2016
Seems to differ from A271061 only at n=1. - R. J. Mathar, Mar 27 2025

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015
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 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata

Cf. A273334.

CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
code=657; stages=128;
rule=IntegerDigits[code,2,10];
g=2*stages+1; (* Maximum size of grid *)
a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
ca=a;
ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
PrependTo[ca,a];
(* Trim full grid to reflect growth by one cell at each stage *)
k=(Length[ca[[1]]]+1)/2;
ca=Table[Table[Part[ca[[n]][[j]],Range[k+1-n,k-1+n]],{j,k+1-n,k-1+n}],{n,1,k}];
on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
Part[on,2^Range[0,Log[2,stages]]] (* Extract relevant terms *)

Conjectures from Colin Barker, May 20 2016: (Start)
a(n) = 2^(n+2)*(2^n-1) for n>1.
a(n) = 6*a(n-1)-8*a(n-2) for n>3.
G.f.: (1-2*x+32*x^2-32*x^3) / ((1-2*x)*(1-4*x)).
(End)

a(8)-a(15) from Lars Blomberg, Jul 18 2016

cp's OEIS Frontend

A211012 Total area of all squares and rectangles after 2^n stages in the toothpick structure of A139250, assuming the toothpicks have length 2.

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