A159295 - OEIS

%I #18 Sep 08 2022 08:45:43

%S 1,721,1842,4025,7856,14124,23936,38654,60090,90407,132374,189223,

%T 264972,364230,492596,656404,863206,1121449,1441050,1832997,2310024,

%U 2886128,3577352,4401210,5377586,6528059,7876926,9450419,11277860

%N 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.

%H G. C. Greubel, <a href="/A159295/b159295.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3,-8,14,0,-14,8,3,-4,1).

%F 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)).

%F 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

%t 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 *)

%o (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

%o (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

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

%K nonn,easy

%O 1,2

%A _David Pasino_, Apr 09 2009

%E Typo in formula corrected by _David Pasino_, Apr 15 2009