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A211008 Triangle read by rows: T(n,k) = number of squares and rectangles of area 2^(k-1) after n-th stage in the toothpick structure of A139250, n>=1, k>=1, assuming the toothpicks have length 2.

%I A211008 #41 Dec 24 2022 22:24:44
%S A211008 0,0,0,2,0,4,0,4,4,4,8,8,2,8,12,4,8,12,4,12,12,4,16,16,4,16,20,4,20,
%T A211008 20,4,32,28,4,40,44,8,2,40,52,12,4,40,52,12,4,44,52,12,4,48,56,12,4,
%U A211008 48,60,12,4,52,60,12,4,64,68,12,4,72,84,16,4
%N A211008 Triangle read by rows: T(n,k) = number of squares and rectangles of area 2^(k-1) after n-th stage in the toothpick structure of A139250, n>=1, k>=1, assuming the toothpicks have length 2.
%C A211008 It appears that the number of rectangles of area 2 in the toothpick structure of A139250 equals the number of hearts in the Q-toothpick cellular automaton of A187210. See conjecture in formula section.
%H A211008 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 A211008 It appears that T(n,2) = A188346(n+2) (checked by hand up to n = 128 in the toothpick structure of A139250).
%e A211008 For n = 8 in the toothpick structure after 8 stages we have that:
%e A211008 T(8,1) = 8 is the number of squares of size 1 X 1.
%e A211008 T(8,2) = 12 is the number of rectangles of size 1 X 2.
%e A211008 T(8,3) = 4 is the number of squares of size 2 X 2.
%e A211008 Written as an irregular array the sequence begins:
%e A211008    0;
%e A211008    0;
%e A211008    0,  2;
%e A211008    0,  4;
%e A211008    0,  4;
%e A211008    4,  4;
%e A211008    8,  8,  2;
%e A211008    8, 12,  4;
%e A211008    8, 12,  4;
%e A211008   12, 12,  4;
%e A211008   16, 16,  4;
%e A211008   16, 20,  4;
%e A211008   20, 20,  4;
%e A211008   32, 28,  4;
%e A211008   40, 44,  8,  2;
%e A211008   40, 52, 12,  4;
%Y A211008 Zero together with the row sums gives A160124.
%Y A211008 Cf. A139250, A159786, A160125, A168131, A187210, A188346, A211016, A211017, A211019.
%K A211008 nonn,tabf
%O A211008 1,4
%A A211008 _Omar E. Pol_, Sep 18 2012