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%I #19 Dec 24 2022 22:24:52
%S 0,0,8,8,24,16,40,104,48,32,168,424,208,96,64,680,1704,848,416,192,
%T 128,2728,6824,3408,1696,832,384,256,10920,27304,13648,6816,3392,1664,
%U 768,512,43688,109924,54608,27296,13632,6784,3328,1536,1024
%N T(n,k) = total area of all squares and rectangles of area 2^(k-1) after 2^n stages in the toothpick structure of A139250, n>=1, k>=1, assuming the toothpicks have length 2. Triangle read by rows.
%C All internal regions in the toothpick structure are squares and rectangles. The area of every internal region is a power of 2.
%H 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>
%F T(n,k) = A211016(n,k)*2^(k-1).
%F T(n,1) = 4*A020988(n-2), n>=2.
%e For n = 5 in the toothpick structure after 2^5 stages we have that:
%e T(5,1) = 168 is the total area of all squares of size 1 X 1.
%e T(5,2) = 424 is the total area of all rectangles of size 1 X 2.
%e T(5,3) = 208 is the total area of all squares of size 2 X 2 and of all rectangles of size 1 X 4.
%e T(5,4) = 96 is the total area of all rectangles of size 2 X 4.
%e T(5,5) = 64 is the total area of all rectangles of size 2 X 8.
%e Triangle begins:
%e 0;
%e 0, 8;
%e 8, 24, 16;
%e 40, 104, 48, 32;
%e 168, 424, 208, 96, 64;
%e 680, 1704, 848, 416, 192, 128;
%e 2728, 6824, 3408, 1696, 832, 384, 256;
%e 10920, 27304, 13648, 6816, 3392, 1664, 768, 512;
%Y Row sums give A211012.
%Y Cf. A020988, A139250, A160124, A211016, A211018, A211019.
%K nonn,tabl
%O 1,3
%A _Omar E. Pol_, Sep 21 2012
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