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A211016 Triangle read by rows: T(n,k) = number of 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.

%I A211016 #22 Dec 24 2022 22:24:49
%S A211016 0,0,4,8,12,4,40,52,12,4,168,212,52,12,4,680,852,212,52,12,4,2728,
%T A211016 3412,852,212,52,12,4,10920,13652,3412,852,212,52,12,4,43688,54612,
%U A211016 13652,3412,852,212,52,12,4,174760,218452,54612,13652,3412,852,212,52,12,4
%N A211016 Triangle read by rows: T(n,k) = number of 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.
%C A211016 All internal regions in the toothpick structure are squares and rectangles.
%H A211016 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 A211016 T(n,k) = A211008(2^n,k) = 4*A211019(n,k).
%F A211016 T(n,1) = 4*A020988(n-2), n>=2.
%e A211016 For n = 5 in the toothpick structure after 2^5 stages we have that:
%e A211016 T(5,1) = 168 is the number of squares of size 1 X 1.
%e A211016 T(5,2) = 212 is the number of rectangles of size 1 X 2.
%e A211016 T(5,3) = 52 is the total number of squares of size 2 X 2 and of rectangles of size 1 X 4.
%e A211016 T(5,4) = 12 is the number of rectangles of size 2 X 4.
%e A211016 T(5,5) = 4 is the number of rectangles of size 2 X 8.
%e A211016 Triangle begins:
%e A211016        0;
%e A211016        0,      4;
%e A211016        8,     12,     4;
%e A211016       40,     52,    12,     4;
%e A211016      168,    212,    52,    12,    4;
%e A211016      680,    852,   212,    52,   12,   4;
%e A211016     2728,   3412,   852,   212,   52,  12,   4;
%e A211016    10920,  13652,  3412,   852,  212,  52,  12,  4;
%e A211016    43688,  54612, 13652,  3412,  852, 212,  52, 12,  4;
%e A211016   174760, 218452, 54612, 13652, 3412, 852, 212, 52, 12, 4;
%Y A211016 Row sums give 0 together with A145655.
%Y A211016 Cf. A020988, A072197,  A080675, A139250, A160124, A211008, A211017, A211018, A211019.
%K A211016 nonn,tabl
%O A211016 1,3
%A A211016 _Omar E. Pol_, Sep 18 2012