A159295 Number of ways that a tile in the form of a strip of n congruent regular hexagons stuck together on successive parallel edges can be surrounded by one layer of copies of itself in a plane. Ways that differ by rotation or reflection are not counted as different. The surrounded tile is the exact surrounded region.

1, 721, 1842, 4025, 7856, 14124, 23936, 38654, 60090, 90407, 132374, 189223, 264972, 364230, 492596, 656404, 863206, 1121449, 1441050, 1832997, 2310024, 2886128, 3577352, 4401210, 5377586, 6528059, 7876926, 9450419, 11277860

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 (4,-3,-8,14,0,-14,8,3,-4,1).

Formula

a(1) = 1, a(2) = 721, and if n > 2 then a(n) = (1/144)*(n^6 + 30*n^5 + 463*n^4 + 3132*n^3 + 11506*n^2 + 10716*n - 1152 + (n odd)(9*n^2 + 90*n + 261)).

G.f.: x*(28*x^11 -285*x^10 +784*x^9 -307*x^8 -1866*x^7 +2566*x^6 +583*x^5 -3036*x^4 +1172*x^3 +1039*x^2 -717*x-1) / ((x-1)^7*(x+1)^3). - Colin Barker, Nov 26 2012

Mathematica

Join[{1, 721}, LinearRecurrence[{4, -3, -8, 14, 0, -14, 8, 3, -4, 1}, {1842, 4025, 7856, 14124, 23936, 38654, 60090, 90407, 132374, 189223}, 30]] (* Harvey P. Dale, Dec 04 2014 *)

Prog

(PARI) x='x+O('x^30); Vec(x*(28*x^11 -285*x^10 +784*x^9 -307*x^8 -1866*x^7 +2566*x^6 +583*x^5 -3036*x^4 +1172*x^3 +1039*x^2 -717*x-1)/( (x-1)^7*(x+1)^3)) \\ G. C. Greubel, Jun 27 2018
(Magma) I:=[1842, 4025, 7856, 14124, 23936, 38654, 60090, 90407, 132374, 189223]; [1, 721] cat [n le 10 select I[n] else 4*Self(n-1) -3*Self(n-2) -8*Self(n-3) +14*Self(n-4) -14*Self(n-6) +8*Self(n-7) +3*Self(n-8) -4*Self(n-9) +Self(n-10): n in [1..30]]; // G. C. Greubel, Jun 27 2018

Crossrefs

Cf. A159294 for analogous problem for strip-of-squares tile.

Sequence in context: A014440 A345558 A345812 * A154515 A241961 A318527

Adjacent sequences: A159292 A159293 A159294 * A159296 A159297 A159298

Keyword

nonn,easy

Author

David Pasino, Apr 09 2009

Extensions

Typo in formula corrected by David Pasino, Apr 15 2009

Status

approved