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

%I A211012 #32 Mar 12 2025 20:16:59
%S A211012 0,0,8,48,224,960,3968,16128,65024,261120,1046528,4190208,16769024,
%T A211012 67092480,268402688,1073676288,4294836224,17179607040,68718952448,
%U A211012 274876858368,1099509530624,4398042316800,17592177655808,70368727400448,281474943156224
%N A211012 Total area of all squares and rectangles after 2^n stages in the toothpick structure of A139250, assuming the toothpicks have length 2.
%C A211012 All internal regions in the toothpick structure are squares and rectangles. The area of every internal region is a power of 2.
%C A211012 Similar to A271061. - _Robert Price_, Mar 30 2016
%C A211012 For n=3,5,..., also the number of minimum vertex colorings in the n-sunlet graph. - _Eric W. Weisstein_, Mar 03 2024
%H A211012 Colin Barker, <a href="/A211012/b211012.txt">Table of n, a(n) for n = 0..1000</a>
%H A211012 N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%H A211012 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MinimumVertexColoring.html">Minimum Vertex Coloring</a>.
%H A211012 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SunletGraph.html">Sunlet Graph</a>.
%H A211012 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-8).
%F A211012 a(n) = 2^n * (2^n-2) = A000079(n)*(A000079(n) - 2) = A159786(2^n) = 8*A006516(n-1), n>=1.
%F A211012 From _Colin Barker_, Mar 30 2016: (Start)
%F A211012 G.f.: 8*x^2 / ((1-2*x)*(1-4*x)).
%F A211012 a(n) = 6*a(n-1)-8*a(n-2) for n>2. (End)
%F A211012 E.g.f.: (1 - exp(2*x))^2. - _Stefano Spezia_, Mar 12 2025
%e A211012 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.
%o A211012 (PARI) concat(vector(2), Vec(8*x^2/((1-2*x)*(1-4*x)) + O(x^50))) \\ _Colin Barker_, Mar 30 2016
%Y A211012 Row sums of triangle A211017, n>=1.
%Y A211012 Cf. A000079, A006516, A139250, A159786, A160124, A211008, A211016, A211018.
%K A211012 nonn,easy
%O A211012 0,3
%A A211012 _Omar E. Pol_, Sep 21 2012