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%I #49 Oct 13 2023 16:14:32
%S 8,24,62,148,338,752,1646,3564,7658,16360,34790,73700,155618,327648,
%T 688094,1441756,3014618,6291416,13107158,27262932,56623058,117440464,
%U 243269582,503316428,1040187338,2147483592,4429184966
%N Number of tilings of the d-dimensional zonotope constructed from d+3 vectors.
%C The zonotope Z(D,d) is the projection of the D-dimensional hypercube onto the d-dimensional space and the tiles are the projections of the d-dimensional faces of the hypercube. Here the codimension, i.e., D-d, is constant = 3 and d >= 0.
%C Also the number of signotopes on r+2 elements of rank r. A signotope on n elements of rank r is a mapping X:{{1..n} choose r}->{+,-} such that for any r+1 indices I={i_0,...,i_r} with i_0 < i_1 < ... < i_r, the sequence X(I-i_0), X(I-i_1), ..., X(I-i_r) changes its sign at most once (see Felsner-Weil reference). - _Manfred Scheucher_, Feb 09 2022
%D A. Bjorner, M. Las Vergnas, B. Sturmfels, N. White and G. M. Ziegler, Oriented Matroids, Encyclopedia of Mathematics 46, Second Edition, Cambridge University Press, 1999.
%D Victor Reiner, The generalized Baues problem, in New Perspectives in Algebraic Combinatorics (Berkeley, CA, 1996-1997), 293-336, Math. Sci. Res. Inst. Publ., 38, Cambridge Univ. Press, Cambridge, 1999.
%H N. Destainville, R. Mosseri and F. Bailly, <a href="https://arxiv.org/abs/cond-mat/0004145">Fixed-boundary octagonal random tilings: a combinatorial approach</a>, arXiv:cond-mat/0004145 [cond-mat.stat-mech], 2000.
%H N. Destainville, R. Mosseri and F. Bailly, <a href="https://doi.org/10.1023/A:1026564710037">Fixed-boundary octagonal random tilings: a combinatorial approach</a>, Journal of Statistical Physics, 102 (2001), no. 1-2, 147-190.
%H S. Felsner and H. Weil, <a href="http://doi.org/10.1016/S0166-218X(00)00232-8">Sweeps, arrangements and signotopes</a>, Discrete Applied Mathematics, Volume 109, Issues 1-2, 2001, Pages 67-94.
%H M. Latapy, <a href="https://arxiv.org/abs/math/0008022">Generalized Integer Partitions, Tilings of Zonotopes and Lattices</a>, arXiv:math/0008022 [math.CO], 2000.
%H G. M. Ziegler, <a href="https://www.mi.fu-berlin.de/math/groups/discgeom/ziegler/Preprintfiles/025PREPRINT.pdf">Higher Bruhat Orders and Cyclic Hyperplane Arrangements</a>, Topology, Volume 32, 1993.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,-13,12,-4).
%F Conjectures from _Colin Barker_, Feb 20 2013: (Start)
%F a(n) = 2*(-3+7*2^n+(-1+2^n)*n).
%F G.f.: -2*(4*x^3-11*x^2+12*x-4) / ((x-1)^2*(2*x-1)^2). (End)
%F The above conjectures are correct; see Proposition 7.1 in Ziegler's article. - _Manfred Scheucher_, Feb 09 2022
%F a(n) = 2 * A133546(n+2). - _Alois P. Heinz_, Feb 11 2022
%e For any Z(D,d), the number of codimension 0 tilings is always 1, with codimension 1 it is 2, with codimension 2 it is 2.D.
%t LinearRecurrence[{6,-13,12,-4},{8,24,62,148},30] (* _Harvey P. Dale_, Oct 13 2023 *)
%o (Python) print([2**(n + 1)*(n + 7) - 2*n - 6 for n in range(100)])
%Y Cf. A006245 (two-dimensional tilings), A060595-A060601. A diagonal of A060637. See also A351383 and A351384 for other diagonals.
%Y Cf. A133546.
%K nonn,nice,easy
%O 0,1
%A Matthieu Latapy (latapy(AT)liafa.jussieu.fr), Apr 12 2001
%E Edited by _Manfred Scheucher_, Mar 08 2022
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