cp's OEIS Frontend

A006357 Number of distributive lattices; also number of paths with n turns when light is reflected from 4 glass plates.

%I A006357 M3396 #76 Jul 02 2025 16:01:54
%S A006357 1,4,10,30,85,246,707,2037,5864,16886,48620,139997,403104,1160693,
%T A006357 3342081,9623140,27708726,79784098,229729153,661478734,1904652103,
%U A006357 5484227157,15791202736,45468956106,130922641160,376976720745,1085461206128,3125460977225
%N A006357 Number of distributive lattices; also number of paths with n turns when light is reflected from 4 glass plates.
%C A006357 Let M denotes the 4 X 4 matrix = row by row (1,1,1,1)(1,1,1,0)(1,1,0,0)(1,0,0,0) and A(n) the vector (x(n),y(n),z(n),t(n))=M^n*A where A is the vector (1,1,1,1) then a(n)=x(n). - _Benoit Cloitre_, Apr 02 2002
%C A006357 In general, the g.f. for p glass plates is A(x) = F_{p-1}(-x)/F_p(x) where F_p(x) = Sum_{k=0,p} (-1)^[(k+1)/2]*C([(p+k)/2],k)*x^k. - _Paul D. Hanna_, Feb 06 2006
%C A006357 a(n)/a(n-1) tends to 2.879385..., the longest diagonal of a nonagon with edge 1; or: sin(4*Pi/9)/sin(Pi/9). The sequence is the INVERT transform of (1, 3, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...). - _Gary W. Adamson_, Jul 16 2015
%D A006357 J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124.
%D A006357 S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 120).
%D A006357 J. Haubrich, Multinacci Rijen [Multinacci sequences], Euclides (Netherlands), Vol. 74, Issue 4, 1998, pp. 131-133.
%D A006357 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006357 T. D. Noe, <a href="/A006357/b006357.txt">Table of n, a(n) for n = 0..200</a>
%H A006357 J. Berman and P. Koehler, <a href="/A006356/a006356.pdf">Cardinalities of finite distributive lattices</a>, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]
%H A006357 Emma L. L. Gao, Sergey Kitaev, and Philip B. Zhang, <a href="https://arxiv.org/abs/1505.04078">Pattern-avoiding alternating words</a>, arXiv:1505.04078 [math.CO], 2015.
%H A006357 Manfred Goebel, <a href="http://dx.doi.org/10.1007/s002000050118">Rewriting Techniques and Degree Bounds for Higher Order Symmetric Polynomials</a>, Applicable Algebra in Engineering, Communication and Computing (AAECC), Volume 9, Issue 6 (1999), 559-573.
%H A006357 G. Kreweras, <a href="http://www.numdam.org/item?id=MSH_1976__53__5_0">Les préordres totaux compatibles avec un ordre partiel</a>, Math. Sci. Humaines No. 53 (1976), 5-30.
%H A006357 G. Kreweras, <a href="/A019538/a019538.pdf">Les préordres totaux compatibles avec un ordre partiel</a>, Math. Sci. Humaines No. 53 (1976), 5-30. (Annotated scanned copy)
%H A006357 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A006357 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A006357 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,3,-1,-1).
%F A006357 G.f.: (1 + 2*x - x^2 - x^3)/( (1 +x)*(1 -3*x +x^3) ). - _Simon Plouffe_ in his 1992 dissertation
%F A006357 a(n) = 2*a(n-1) + 3*a(n-2) - a(n-3) - a(n-4).
%F A006357 a(n) is asymptotic to z(4)*w(4)^n where w(4) = (1/2)/cos(4*Pi/9) and z(4) is the root 1 < x < 2 of P(4, X) = 1 + 27*X - 324*X^2 + 243*X^3. - _Benoit Cloitre_, Oct 16 2002
%F A006357 Binomial transform of A122167(unsigned): (1, 3, 3, 11, 10, 40, 33, 146, ...). - _Gary W. Adamson_, Nov 24 2007
%F A006357 G.f.: 1/(-x-1/(-x-1/(-x-1/(-x-1)))). - _Paul Barry_, Mar 24 2010
%t A006357 LinearRecurrence[{2,3,-1,-1},{1,4,10,30},30] (* _Harvey P. Dale_, Nov 18 2013 *)
%o A006357 (PARI) a(n)=local(p=4);polcoeff(sum(k=0,p-1,(-1)^((k+1)\2)*binomial((p+k-1)\2,k)* (-x)^k)/sum(k=0,p,(-1)^((k+1)\2)*binomial((p+k)\2,k)*x^k+x*O(x^n)),n) \\ _Paul D. Hanna_
%Y A006357 Cf. A000217, A000330, A050446, A050447, A006356-A006359, A025030, A030112-A030116, A122167, A091024.
%Y A006357 Cf. A038197 (4-wave sequence).
%K A006357 nonn,nice,easy
%O A006357 0,2
%A A006357 _N. J. A. Sloane_
%E A006357 Recurrence, alternative description from Jacques Haubrich (jhaubrich(AT)freeler.nl)
%E A006357 More terms from _James Sellers_, Dec 24 1999
%E A006357 More terms from _Paul D. Hanna_, Feb 06 2006