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

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

Crossrefs

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

Programs

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

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 *)

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

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

Extensions

Typo in formula corrected by David Pasino, Apr 15 2009