A354012 Number of tilings of a 4 X n rectangle using 2 X 2 tiles, right trominoes and dominoes.

1, 1, 17, 81, 702, 4623, 35044, 248045, 1819731, 13110984, 95362462, 690253391, 5008926698, 36300216768, 263252448712, 1908449014617, 13837881924141, 100326715619679, 727420462629671, 5274035027493046, 38238994112367061, 277246970248002472, 2010151423463689959

Offset

0,3

Comments

For tiling algorithm, see A351322.

Links

Table of n, a(n) for n=0..22.

Index entries for linear recurrences with constant coefficients, signature (5,28,-69,-142,194,78,-57,-36,70,-32).

Formula

G.f.: (1 - 4*x - 16*x^2 + 37*x^3 + 32*x^4 - 34*x^5 + 4*x^6 + 2*x^7 - 2*x^8) / (1 - 5*x - 28*x^2 + 69*x^3 + 142*x^4 - 194*x^5 - 78*x^6 + 57*x^7 + 36*x^8 - 70*x^9 + 32*x^10).

a(n)=5*a(n-1) + 28*a(n-2) - 69*a(n-3) - 142*a(n-4) + 194*a(n-5) + 78*a(n-6) - 57*a(n-7) - 36*a(n-8) + 70*a(n-9) - 32*a(n-10).

Example

a(2)=17, mirroring included (h: horizontal, v: vertical):
    v     v          h,v                      v           h
   ___   ___   ___   ___   ___   ___   ___   ___   ___   ___
  |   | |   | |   | |___| |___| | | | |___| |___| |___| |  _|
  |___| |___| |___| |_  | |___| |_|_| | | | |___| |   | |_| |
  |___| | | | |   | | |_| |___| | | | |_|_| | | | |___| | |_|
  |___| |_|_| |___| |___| |___| |_|_| |___| |_|_| |___| |___|
    2  +  2  +  1  +  4  +  1  +  1  +  1  +  2  +  1  +  2  = 17.

Prog

(Maxima) See A352589.

Crossrefs

Cf. A351322, A352589, A354010, A354011.

Sequence in context: A044585 A347501 A197397 * A053826 A351267 A184982

Adjacent sequences: A354009 A354010 A354011 * A354013 A354014 A354015

Keyword

nonn,easy

Author

Gerhard Kirchner, May 14 2022

Status

approved