A211008 - OEIS

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 #41 Dec 24 2022 22:24:44

%S 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 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 48,60,12,4,52,60,12,4,64,68,12,4,72,84,16,4

%N 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 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 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 It appears that T(n,2) = A188346(n+2) (checked by hand up to n = 128 in the toothpick structure of A139250).

%e For n = 8 in the toothpick structure after 8 stages we have that:

%e T(8,1) = 8 is the number of squares of size 1 X 1.

%e T(8,2) = 12 is the number of rectangles of size 1 X 2.

%e T(8,3) = 4 is the number of squares of size 2 X 2.

%e Written as an irregular array the sequence begins:

%e 0;

%e 0, 2;

%e 0, 4;

%e 4, 4;

%e 8, 8, 2;

%e 8, 12, 4;

%e 12, 12, 4;

%e 16, 16, 4;

%e 16, 20, 4;

%e 20, 20, 4;

%e 32, 28, 4;

%e 40, 44, 8, 2;

%e 40, 52, 12, 4;

%Y Zero together with the row sums gives A160124.

%Y Cf. A139250, A159786, A160125, A168131, A187210, A188346, A211016, A211017, A211019.

%K nonn,tabf

%O 1,4

%A _Omar E. Pol_, Sep 18 2012