A159294 Number of ways that a 1 X n rectangular tile T, marked into n unit squares, can be surrounded by one layer of copies of itself laid in the plane grid generated by the units of T. Ways that differ by rotation or reflection are not counted as different. The surrounded tile is the exact surrounded region.

1, 153, 301, 517, 825, 1234, 1774, 2454, 3310, 4351, 5619, 7123, 8911, 10992, 13420, 16204, 19404, 23029, 27145, 31761, 36949, 42718, 49146, 56242, 64090, 72699, 82159, 92479, 103755, 115996, 129304, 143688, 159256, 176017, 194085, 213469, 234289, 256554, 280390, 305806, 332934

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).

David Pasino, Regular Polygon Rug and Bar Tiles

Formula

For n>1, a(n) = (1/16)*(n^4 + 30*n^3 + 246*n^2 + 476*n + 256 + (1 if n odd, 0 if n even)*(6*n + 9)).

G.f.: -x*(63*x^7-173*x^6+15*x^5+335*x^4-228*x^3-157*x^2+150*x+1) / ((x-1)^5*(x+1)^2). - Colin Barker, Nov 26 2012

Mathematica

Join[{1, 153}, LinearRecurrence[{3, -1, -5, 5, 1, -3, 1}, {301, 517, 825, 1234, 1774, 2454, 3310}, 49]] (* G. C. Greubel, Jun 27 2018 *)

Prog

(PARI) my(x='x+O('x^30)); Vec(-x*(63*x^7-173*x^6+15*x^5 +335*x^4 -228*x^3 - 157*x^2+150*x+1)/((x-1)^5*(x+1)^2)) \\ G. C. Greubel, Jun 27 2018
(Magma) I:=[301, 517, 825, 1234, 1774, 2454, 3310]; [1, 153] cat [n le 7 select I[n] else 3*Self(n-1) -Self(n-2) -5*Self(n-3) +5*Self(n-4) + Self(n-5) -3*Self(n-6) +Self(n-7): n in [1..30]]; // G. C. Greubel, Jun 27 2018

Crossrefs

Cf. A159295 for analogous problem for strip-of-hexagons tile.

Sequence in context: A194660 A389842 A348938 * A332228 A349755 A387162

Adjacent sequences: A159291 A159292 A159293 * A159295 A159296 A159297

Keyword

nonn,easy

Author

David Pasino, Apr 09 2009

Status

approved