A359021 Number of inequivalent tilings of a 5 X n rectangle using integer-sided square tiles.

1, 1, 5, 10, 39, 77, 521, 1985, 8038, 32097, 130125, 525676, 2131557, 8635656, 35017970, 141968455, 575692056, 2334344849, 9465939422, 38384559168, 155652202456, 631178976378, 2559476952229, 10378857744374, 42087027204278, 170665938023137, 692062856184512

Offset: 0

Author

John Mason, Dec 12 2022

Keywords

• nonn

Examples

			a(2) is 5 because of:
  +-+-+ +-+-+ +-+-+ +-+-+ +-+-+
  | | | |   | |   | |   | |   |
  +-+-+ +-+-+ +   + +   + +-+-+
  | | | |   | |   | |   | |   |
  +-+-+ +   + +-+-+ +-+-+ +   +
  | | | |   | |   | | | | |   |
  +-+-+ +-+-+ +-+-+ +-+-+ +-+-+
  | | | |   | |   | | | | | | |
  +-+-+ +   + +   + +-+-+ +-+-+
  | | | |   | |   | | | | | | |
  +-+-+ +-+-+ +-+-+ +-+-+ +-+-+
		

Links

• John Mason, Table of n, a(n) for n = 0..1000
• John Mason, Counting tilings of width 5 rectangles

Crossrefs

Column k = 5 of A227690.

Sequences for fixed and free (inequivalent) tilings of m X n rectangles, for 2 <= m <= 10:

Fixed: A000045, A002478, A054856, A054857, A219925, A219926, A219927, A219928, A219929.

Free: A001224, A359019, A359020, A359021, A359022, A359023, A359024, A359025, A359026.

Cf. A079975.

Formula

For even n > 5:

a(n) = (A054857(n) + A079975(n) + 2*A054857(n/2) + 2* fixed_md(n/2) + 2*A054857((n-4)/2) + 4*A054857((n-2)/2) + 2* (A054857((n/2)-1) + fixed_md((n/2)-1)))/4.

For odd n > 5:

a(n) = (A054857(n) + A079975(n) + 2*A054857((n-1)/2) + 4*A054857((n-3)/2) + 2*fixed_md((n-3)/2) + 2*A054857((n-5)/2) + 2*fixed_md((n-1)/2))/4.

where

fixed_md(1)=1, fixed_md(2)=3, fixed_md(3)=15 and for n > 3, fixed_md(n) = A054857(n-1) + A054857(n-2) + fixed_md(n-2)+ fixed_md(n-1) + 2*A054857(n-3) + fixed_md(n-3).