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A381555 Triangle read by rows T(n,k) is the number of diamond coverings for a specific number of diamonds covering an even length row of triangles.

%I A381555 #29 Mar 06 2025 10:55:56
%S A381555 1,4,1,8,4,1,13,16,4,1,19,41,24,4,1,26,85,85,32,4,1,34,155,231,145,40,
%T A381555 4,1,43,259,532,489,221,48,4,1,53,406,1092,1365,891,313,56,4,1,64,606,
%U A381555 2058,3333,2926,1469,421,64,4,1,76,870,3630,7359,8294,5551,2255,545,72,4
%N A381555 Triangle read by rows T(n,k) is the number of diamond coverings for a specific number of diamonds covering an even length row of triangles.
%C A381555 The total number of ways the diamond can cover a single row of length(n) triangles is the Fibonacci series. This total can be subdivided into categories based on the number of covering diamonds. The number of categories increase with the length of the row providing the structure of the triangle (see illustrations in the link below).
%C A381555 The above process provides a way to subdivide the individual Fibonacci numbers.
%C A381555 Comparing the diamond covering of a row of triangles shown here to the diamond corona of a hexagon A380346 or a diamond A380416 may be instructive.
%C A381555 A381552 provides additional graphics that help explain the diamond covering.
%H A381555 Craig Knecht, <a href="/A381555/a381555.png">Diamond covering of a even length row of triangles</a>.
%H A381555 Craig Knecht, <a href="/A381555/a381555_3.png">Geometric bisection of the Fibonacci sequence.</a>
%H A381555 Craig Knecht, <a href="/A381555/a381555_1.png">Subdividing the individual Fibonacci numbers</a>.
%H A381555 Walter Trump, <a href="/A381555/a381555_2.png">Diamond covering producing the Fibonacci sequence</a>.
%e A381555 Triangle begins:
%e A381555   1, 4;
%e A381555   1, 8, 4;
%e A381555   1, 13, 16, 4;
%e A381555   1, 19, 41, 24, 4;
%e A381555   1, 26, 85, 85, 32, 4;
%e A381555   1, 34, 155, 231, 145, 40, 4;
%Y A381555 Cf. A063496, A081219, A102083, A323847, A380346, A380416, A381552.
%K A381555 nonn,tabf
%O A381555 0,2
%A A381555 _Craig Knecht_, Feb 27 2025