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

Author

Omar E. Pol, Sep 21 2012

Keywords

• nonn
• easy

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

Examples

			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.
		

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).

Crossrefs

Row sums of triangle A211017, n>=1.

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

Programs

PARI

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

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