A249776 Decimal expansion of the connective constant of the (3.12^2) lattice.

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, 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, 2, 5, 0, 2

Offset: 1

Author

Charles R Greathouse IV, Nov 05 2014

Keywords

• nonn
• cons

Comments

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.

Examples

			1.71104129684484846411708746310445406799321932692481959770080785839492...
		

Links

• Charles R Greathouse IV, Illustration of the (3.12^2) lattice
• I. Jensen and A. J. Guttmann, Self-avoiding walks, neighbour-avoiding walks and trails on semiregular lattices, J. Phys. A: Math. Gen. 31 (1998), pp. 8137-8145.
• Index entries for algebraic numbers, degree 12

Crossrefs

Other connective constants: A179260 (hexagonal or honeycomb lattice).

Programs

Mathematica

(* Illustration of the (3.12^2) lattice. *)
hex312[frac_] := {Re[#], Im[#]} & /@
  Flatten[Table[
    With[{a = Exp[2 Pi I (n - 1/2)/6], b = Exp[2 Pi I ( n + 1/2)/6],
      c = Exp[2 Pi I (n + 3/2)/6]}, {(1 - frac) b +
       frac a, (1 - frac) b + frac c}], {n, 6}]]
shiftPoly[shifts_, coords_] :=
Line[Append[#, #[[1]]]] & /@
  Outer[#1 + #2 &, shifts*1.001, coords, 1, 1]
tri = 1/5; (* Arbitrary, subject to 0 < tri < 1/2; determines size of triangles compared to hexagons. *)
Graphics[{Gray,
  shiftPoly[{{0, 0}, {Sqrt[3], 0}, {2 Sqrt[3], 0}, {3 Sqrt[3],
     0}, {Sqrt[3]/2, 3/2}, {3 Sqrt[3]/2, 3/2}, {5 Sqrt[3]/2,
     3/2}, {7 Sqrt[3]/2, 3/2}, {0, 3}, {Sqrt[3], 3}, {2 Sqrt[3],
     3}, {3 Sqrt[3], 3}, {Sqrt[3]/2, 9/2}, {3 Sqrt[3]/2,
     9/2}, {5 Sqrt[3]/2, 9/2}, {7 Sqrt[3]/2, 9/2}}, hex312[tri]]}]

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]