Aarnav Gogri - cp's OEIS frontend

User: Aarnav Gogri

Aarnav Gogri has authored 2 sequences.

A356623 Number of ways to tile a hexagonal strip made up of 4*n+2 equilateral triangles, using triangles and diamonds.

2, 18, 148, 1208, 9854, 80378, 655632, 5347896, 43622018, 355818522, 2902360468, 23674136576, 193106524430, 1575142124306, 12848207584320, 104800979913168, 854846508252578, 6972859922465346, 56876614724333236

Offset: 0

Greg Dresden and Aarnav Gogri, Aug 17 2022

nonn

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    \_\.

			For n=3, here is one of the a(3)=1208 ways to tile this strip (of 14 triangles) using triangles and diamonds.
    ____________
   /\  /\   \   \
  /__\/  \___\ __\
  \  /\  /   /\  /
   \/__\/__ /__\/

Index entries for linear recurrences with constant coefficients, signature (9, -7, 1).

Cf. A356622, A190984.

LinearRecurrence[{9, -7, 1}, {2, 18, 148}, 40]

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

A356622 Number of ways to tile a hexagonal strip made up of 4*n equilateral triangles, using triangles and diamonds.

Original entry on oeis.org

1, 5, 39, 317, 2585, 21085, 171987, 1402873, 11443033, 93339173, 761354199, 6210256613, 50656169297, 413195081581, 3370372805763, 27491645850097, 224245398092113, 1829137434684101, 14920010771362215

Offset: 0

Greg Dresden and Aarnav Gogri, Aug 16 2022

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    \_\.

			For n=4, here is one of the a(4)=2585 ways to tile this strip (of 16 triangles) using triangles and diamonds.
    ________________
   /   /\  /\  /   /
  /__ /  \/__\/__ /
  \  /\  /\   \  /\
   \/__\/__\___\/__\

Index entries for linear recurrences with constant coefficients, signature (9,-7,1).

Bisection of A355327. Cf. A356623.

LinearRecurrence[{9, -7, 1}, {1, 5, 39}, 40]

a(n) = A355327(2*n).
a(n) = 9*a(n-1) - 7*a(n-2) + a(n-3).
G.f.: (1 - 4 x + x^2)/(1 - 9 x + 7 x^2 - x^3).

cp's OEIS Frontend

User: Aarnav Gogri

A356623 Number of ways to tile a hexagonal strip made up of 4*n+2 equilateral triangles, using triangles and diamonds.

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