A375823 Number of ways to tile a 3-row trapezoid of average length n with triangular and rectangular tiles, each of size 3.

0, 1, 3, 6, 16, 43, 107, 271, 695, 1769, 4499, 11464, 29202, 74360, 189382, 482339, 1228417, 3128538, 7967848, 20292665, 51681683, 131623881, 335222157, 853749843, 2174345679, 5537663440, 14103422412, 35918853816, 91478793556, 232979863477, 593357374127

Offset

0,3

Comments

Here is the 3-row trapezoid of average length 6 (with 18 cells):

___ ___ ___ ___ ___

| | | | | |

_|___|___|___|___|_ _|_

| | | | | | |

_|___|___|___|___|_ _|___|_

| | | | | | | |

|___|___|___|___|___|___|___|,

and here are the two types of (triangular and rectangular) tiles of size 3, which can be rotated as needed:

___

| |

_|___|_ ___________

| | | | | | |

|___|___|, |___|___|___|.

As an example, here is one of the a(6) = 107 ways to tile the 3-row trapezoid

___ ___ ___________

| | | |

_| _|_ |___________|_

| | | | | |

_| _| |_ |_ _| |_

| | | | | |

|___|_______|___|___|_______|.

Links

Table of n, a(n) for n=0..30.

Index entries for linear recurrences with constant coefficients, signature (2,0,4,-1,0,-1).

Formula

a(n) = 2*a(n-1) + 4*a(n-3) - a(n-4) - a(n-6).

G.f.: x*(1 + x)/((1 + x^2 - x^3)*(1 - 2*x - x^2 - x^3)).

a(n) = (A077939(n) - A077961(n))/2.

Mathematica

LinearRecurrence[{2, 0, 4, -1, 0, -1}, {0, 1, 3, 6, 16, 43}, 40]

Crossrefs

Cf. A077939, A077961, A375821.

Sequence in context: A218207 A019497 A091488 * A202839 A371705 A007561

Adjacent sequences: A375820 A375821 A375822 * A375824 A375825 A375826

Keyword

nonn,easy

Author

Greg Dresden and Mingjun Oliver Ouyang, Aug 30 2024

Status

approved