%I A006356 M2578
%S A006356 1,3,6,14,31,70,157,353,793,1782,4004,8997,20216,45425,102069,229347,
%T A006356 515338,1157954,2601899,5846414,13136773,29518061,66326481,149034250,
%U A006356 334876920,752461609,1690765888,3799116465,8536537209,19181424995
%N A006356 a(n)=2*a(n-1)+a(n-2)-a(n-3).
%C A006356 Number of distributive lattices; also number of paths with n turns when 
               light is reflected from 3 glass plates.
%C A006356 Let u(k), v(k), w(k) be be defined by u(1)=1, v(1)=0, w(1)=0 and u(k+1)=u(k)+v(k)+w(k), 
               v(k+1)=u(k)+v(k), w(k+1)=u(k); then {u(n)} = 1,1,3,6,14,31,... (this 
               sequence with an extra initial 1), {v(n)} = 0,1,2,5,11,25,... (A006054 
               with its initial 0 deleted) and {w(n)} = {u(n)} prefixed by an extra 
               0 = A077998 with an extra initial 0. - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Apr 05 2002. Also u(k)^2+v(k)^2+w(k)^2 = u(2k). - Gary Adamson, Dec 
               23 2003.
%C A006356 The n-th term of the series is the number of paths for a ray of light 
               that enters two layers of glass and then is reflected exactly n times 
               before leaving the layers of glass.
%C A006356 One such path (with 2 plates of glass and 3 reflections) might be:
%C A006356 ...\........./..................
%C A006356 --------------------------------
%C A006356 ....\/\..../....................
%C A006356 --------------------------------
%C A006356 ........\/......................
%C A006356 --------------------------------
%C A006356 For a k-glass sequence, say a(n,k), a(n,k) is always asymptotic to z(k)*w(k)^n 
               where w(k)=(1/2)/cos(k*Pi/(2k+1)) and it is conjectured that z(k) 
               is the root 1<x<2 of a polynomial of degree Phi(2k+1)/2
%C A006356 Number of ternary sequences of length n-1 such that every pair of consecutive 
               digits has a sum less than 3. That is to say, the pairs (1,2), (2,
               1) and (2,2) do not appear. - George J. Schaeffer (gschaeff(AT)andrew.cmu.edu), 
               Sep 07 2004
%C A006356 Number of weakly up-down sequences of length n using the digits {1,2,
               3}. When n=2 the sequences are 11, 12, 13, 22, 23, 33.
%C A006356 Form the graph with matrix A=[1, 1, 1; 1, 0, 0; 1, 0, 1]. Then A006356 
               counts walks of length n that start at the degree 4 vertex. - Paul 
               Barry (pbarry(AT)wit.ie), Oct 02 2004
%C A006356 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 (pauldhanna(AT)juno.com), Feb 06 2006
%C A006356 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 27 2009: 
               (Start)
%C A006356 Equals the INVERT transform of (1, 2, 1, 1, 1,...) equivalent to a(n) 
               =
%C A006356 a(n-1) + 2*a(n-2) + a(n-3) + a(n-4) + ... + 1. a(6) = 70 = (31 + 2*14 
               + 6 + 3 + 1 + 1)
%C A006356 (End)
%D A006356 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A006356 S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques 
               Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%D A006356 J. Berman and P. Koehler, Cardinalities of finite distributive lattices, 
               Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 
               103-124.
%D A006356 S. J. Cyvin and I. Gutman, Kekule structures in benzenoid hydrocarbons, 
               Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see 
               p. 120).
%D A006356 Manfred Goebel, Rewriting Techniques and Degree Bounds for Higher Order 
               Symmetric Polynomials, Applicable Algebra in Engineering, Communication 
               and Computing (AAECC), Volume 9, Issue 6 (1999), 559-573.
%D A006356 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, 
               Reading, MA, 2nd edition, p. 291 (very briefly without generalizations).
%D A006356 J. Haubrich, Multinacci Rijen [Multinacci sequences], Euclides (Netherlands), 
               Vol. 74, Issue 4, 1998, pp. 131-133.
%D A006356 V. E. Hoggatt Jr. and M. Bicknell-Johnson, Reflections across two and 
               three glass plates, Fibonacci Quarterly, volume 17 (1979), 118-142.
%D A006356 B. Junge and V. E. Hoggatt, Jr., Polynomials arising from reflections 
               across multiple plates, Fib. Quart., 11 (1973), 285-291.
%D A006356 Jay Kappraff, Beyond Measure, A Guided Tour Through Nature, Myth and 
               Number, World Scientific, 2002.
%D A006356 G. Kreweras, Les preordres totaux compatibles avec un ordre partiel. 
               Math. Sci. Humaines No. 53 (1976), 5-30.
%D A006356 Leo Moser, Problem B-6: some reflections, Fib. Quat. Vol. 1, No. 4 (1963), 
               75-76..
%D A006356 L. Moser and M. Wyman, Multiple reflections. Fib. Quart., 11 (1973).
%D A006356 P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), 
               no. 1, 22-31.
%D A006356 Roman Witula, Damian Slota and Adam Warzynski, Quasi-Fibonacci Numbers 
               of the Seventh Order, Journal of Integer Sequences, Vol. 9 (2006), 
               Article 06.4.3.
%H A006356 T. D. Noe, <a href="b006356.txt">Table of n, a(n) for n=0..200</a>
%H A006356 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A006356 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A006356 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A006356 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=451">
               Encyclopedia of Combinatorial Structures 451</a>
%F A006356 a(n) is asymptotic to z(3)*w(3)^n where w(3)=(1/2)/cos(3*Pi/7) and z(3) 
               is the root 1<X<2 of P(3, X) = 1-14*X-49*X^2+49*X^3. w(3)= 2.2469796.... 
               z(3)=1.220410935...
%F A006356 G.f.: A(x) = (1+x-x^2)/(1-2*x-x^2+x^3). - Paul D. Hanna (pauldhanna(AT)juno.com), 
               Feb 06 2006
%F A006356 a(n) = a(n-1) + a(n-2) + A006054(n+1). - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Jun 05 2008
%p A006356 A006356:=-(-1-z+z**2)/(1-2*z-z**2+z**3); [Conjectured by S. Plouffe in 
               his 1992 dissertation.]
%o A006356 (PARI) {a(n)=local(p=3);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)} (Hanna)
%Y A006356 Cf. A000217, A000330, A050446, A050447, A006054, A077998, A052534, A052994, 
               A052949.
%Y A006356 See also A006357-A006359, A025030, A030112-A030116.
%Y A006356 Cf. A038196 (3-wave sequence).
%Y A006356 Sequence in context: A091601 A063119 A106803 this_sequence A077998 A090165 
               A129954
%Y A006356 Adjacent sequences: A006353 A006354 A006355 this_sequence A006357 A006358 
               A006359
%K A006356 nonn,easy,nice
%O A006356 0,2
%A A006356 N. J. A. Sloane (njas(AT)research.att.com).
%E A006356 Recurrence, alternative description from Jacques Haubrich (jhaubrich(AT)freeler.nl).
%E A006356 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Dec 24 1999
%E A006356 Added an alternative definition. - Andy Niedermaier (aniederm(AT)math.ucsd.edu), 
               Nov 11 2008