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, 1125899839733760

Offset

0,3

Comments

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

Links

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).

Formula

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

Example

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.

Mathematica

A211012[n_] := If[n == 0, 0, #*(#-2) & [2^n]]; Array[A211012, 30, 0] (* Paolo Xausa, May 17 2026 *)
(* Alternative: *)
LinearRecurrence[{6, -8}, {0, 0, 8}, 30] (* Paolo Xausa, May 17 2026 *)

Prog

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

Crossrefs

Row sums of triangle A211017, n>=1.

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

Sequence in context: A261975 A087914 A271061 * A081084 A230931 A073390

Adjacent sequences: A211009 A211010 A211011 * A211013 A211014 A211015

Keyword

nonn,easy

Author

Omar E. Pol, Sep 21 2012

Status

approved