OFFSET
0,4
COMMENTS
Here is the hexagonal strip:
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/\ /\ /\ /\ / \ /
/__\/__\/__\/__\/ ... \/
\ /\ /\ /\ /\ /\
\/__\/__\/__\/__\ /__\
The two types of tiles are singles and triples (each of which can be rotated). Here are the two types of tiles:
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\ / \ /\ /
\/ and \/__\/.
LINKS
Index entries for linear recurrences with constant coefficients, signature (0,0,0,7,0,0,0,1,0,0,0,3,0,0,0,-3,0,0,0,3,0,0,0,-1).
FORMULA
a(n) = 7*a(n-4) + a(n-8) + 3*a(n-12) - 3*a(n-16) + 3*a(n-20) - a(n-24).
a(4*n+2) = b(2*n+1)^2 + 2*Sum_{k=0..n} b(2*(n-k))*(a(4*k)*b(2*(n-k)-1) + a(4*k-1)*b(2*(n-k))), where b(n)=A000930(n), the Narayana's Cows sequence, so long as we agree that a(-1)=0 and b(-1)=0. - Greg Dresden, Sep 16 2023
EXAMPLE
For n=16, here is one of the a(16)=1191 ways to tile this strip (of 16 triangles) using singles and triples.
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/ / /\ /
/ / /__\ __ /
\ /\ /\ /\
\/__\/__\___ /__\
MATHEMATICA
g[-3] = 0; g[-2] = 0; g[-1] = 0; g[0] = 1; g[1] = 1; g[2] = 1; g[3] = 2; g[4] = 3;
g[n_] := g[n] = Switch[Mod[n, 4],
0, g[n - 1] + g[n - 4] + g[n - 5] + g[n - 6] + g[n - 7],
1, g[n - 1] + g[n - 4] + g[n - 6],
2, g[n - 1] + 2 g[n - 3] + g[n - 5] + g[n - 6] + g[n - 7],
3, g[n - 1] + g[n - 3] + g[n - 4] + g[n - 6]];
Table[g[n], {n, 0, 30}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Greg Dresden and Yiwei Wang, Sep 02 2023
STATUS
approved