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%I #12 Aug 21 2023 11:47:49
%S 1,7,1,1,0,4,1,2,9,6,8,4,4,8,4,8,4,6,4,1,1,7,0,8,7,4,6,3,1,0,4,4,5,4,
%T 0,6,7,9,9,3,2,1,9,3,2,6,9,2,4,8,1,9,5,9,7,7,0,0,8,0,7,8,5,8,3,9,4,9,
%U 2,5,0,2
%N Decimal expansion of the connective constant of the (3.12^2) lattice.
%C An algebraic integer of degree 12: largest real root of x^12 - 4x^8 - 8x^7 - 4x^6 + 2x^4 + 8x^3 + 12x^2 + 8x + 2.
%H Charles R Greathouse IV, <a href="/A249776/a249776.png">Illustration of the (3.12^2) lattice</a>
%H I. Jensen and A. J. Guttmann, <a href="http://dx.doi.org/10.1088/0305-4470/31/40/008">Self-avoiding walks, neighbour-avoiding walks and trails on semiregular lattices</a>, J. Phys. A: Math. Gen. 31 (1998), pp. 8137-8145.
%H <a href="/index/Al#algebraic_12">Index entries for algebraic numbers, degree 12</a>
%e 1.71104129684484846411708746310445406799321932692481959770080785839492...
%t (* Illustration of the (3.12^2) lattice. *)
%t hex312[frac_] := {Re[#], Im[#]} & /@
%t Flatten[Table[
%t With[{a = Exp[2 Pi I (n - 1/2)/6], b = Exp[2 Pi I ( n + 1/2)/6],
%t c = Exp[2 Pi I (n + 3/2)/6]}, {(1 - frac) b +
%t frac a, (1 - frac) b + frac c}], {n, 6}]]
%t shiftPoly[shifts_, coords_] :=
%t Line[Append[#, #[[1]]]] & /@
%t Outer[#1 + #2 &, shifts*1.001, coords, 1, 1]
%t tri = 1/5; (* Arbitrary, subject to 0 < tri < 1/2; determines size of triangles compared to hexagons. *)
%t Graphics[{Gray,
%t shiftPoly[{{0, 0}, {Sqrt[3], 0}, {2 Sqrt[3], 0}, {3 Sqrt[3],
%t 0}, {Sqrt[3]/2, 3/2}, {3 Sqrt[3]/2, 3/2}, {5 Sqrt[3]/2,
%t 3/2}, {7 Sqrt[3]/2, 3/2}, {0, 3}, {Sqrt[3], 3}, {2 Sqrt[3],
%t 3}, {3 Sqrt[3], 3}, {Sqrt[3]/2, 9/2}, {3 Sqrt[3]/2,
%t 9/2}, {5 Sqrt[3]/2, 9/2}, {7 Sqrt[3]/2, 9/2}}, hex312[tri]]}]
%o (PARI) polrootsreal(x^12-4*x^8-8*x^7-4*x^6+2*x^4+8*x^3+12*x^2+8*x+2)[4]
%Y Other connective constants: A179260 (hexagonal or honeycomb lattice).
%K nonn,cons
%O 1,2
%A _Charles R Greathouse IV_, Nov 05 2014
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