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

Offset: 1

Author

David Pasino, Apr 09 2009

Keywords

• nonn
• easy

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).

Crossrefs

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

Programs

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

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 *)

PARI

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

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