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A356623
Number of ways to tile a hexagonal strip made up of 4*n+2 equilateral triangles, using triangles and diamonds.
3
2, 18, 148, 1208, 9854, 80378, 655632, 5347896, 43622018, 355818522, 2902360468, 23674136576, 193106524430, 1575142124306, 12848207584320, 104800979913168, 854846508252578, 6972859922465346, 56876614724333236
OFFSET
0,1
COMMENTS
Here is the hexagonal strip:
________________ ____
/\ /\ /\ /\ / \ /\
/__\/__\/__\/__\/ ... \/__\
\ /\ /\ /\ /\ /\ /
\/__\/__\/__\/__\ /__\/
The two types of tiles are triangles and diamonds (each of which can be rotated). Here are the two types of tiles:
____ ____
\ / \ \
\/ and \___\.
FORMULA
a(n) = 9*a(n-1) - 7*a(n-2) + a(n-3).
a(n) = 2^(n+1) + Sum_{k=1..n} 2^(n-k)*(3*b(k) - b(k-1)) for n>=1, for b(n) = A356622(n).
G.f.: 2/(1 - 9*x + 7*x^2 - x^3).
a(n) = 2 + a(n-1) + 2*Sum_{k=1..n}(a(k-1)+A356622(k)). - Aarnav Gogri, Aug 17 2022
a(n+3) = 2*b(n+3) + Sum_{k=0..n} a(k)*b(n-k) for b(n) = A190984(n+1). - Greg Dresden and Aarnav Gogri, Aug 24 2022
EXAMPLE
For n=3, here is one of the a(3)=1208 ways to tile this strip (of 14 triangles) using triangles and diamonds.
____________
/\ /\ \ \
/__\/ \___\ __\
\ /\ / /\ /
\/__\/__ /__\/
MATHEMATICA
LinearRecurrence[{9, -7, 1}, {2, 18, 148}, 40]
CROSSREFS
Sequence in context: A037565 A329170 A125835 * A091170 A207319 A361961
KEYWORD
nonn
AUTHOR
Greg Dresden and Aarnav Gogri, Aug 17 2022
STATUS
approved