0,2

Number of distributive lattices; also number of paths with n turns when light is reflected from 3 glass plates.

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.

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.

One such path (with 2 plates of glass and 3 reflections) might be:

...\........./..................

--------------------------------

....\/\..../....................

--------------------------------

........\/......................

--------------------------------

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

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

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.

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

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

Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 27 2009: (Start)

Equals the INVERT transform of (1, 2, 1, 1, 1,...) equivalent to a(n) =

a(n-1) + 2*a(n-2) + a(n-3) + a(n-4) + ... + 1. a(6) = 70 = (31 + 2*14 + 6 + 3 + 1 + 1)

(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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.

J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124.

S. J. Cyvin and I. Gutman, Kekule structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 120).

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.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd edition, p. 291 (very briefly without generalizations).

J. Haubrich, Multinacci Rijen [Multinacci sequences], Euclides (Netherlands), Vol. 74, Issue 4, 1998, pp. 131-133.

V. E. Hoggatt Jr. and M. Bicknell-Johnson, Reflections across two and three glass plates, Fibonacci Quarterly, volume 17 (1979), 118-142.

B. Junge and V. E. Hoggatt, Jr., Polynomials arising from reflections across multiple plates, Fib. Quart., 11 (1973), 285-291.

Jay Kappraff, Beyond Measure, A Guided Tour Through Nature, Myth and Number, World Scientific, 2002.

G. Kreweras, Les preordres totaux compatibles avec un ordre partiel. Math. Sci. Humaines No. 53 (1976), 5-30.

Leo Moser, Problem B-6: some reflections, Fib. Quat. Vol. 1, No. 4 (1963), 75-76..

L. Moser and M. Wyman, Multiple reflections. Fib. Quart., 11 (1973).

P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.

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.

T. D. Noe, Table of n, a(n) for n=0..200

Index entries for sequences related to linear recurrences with constant coefficients

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.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 451

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...

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

a(n) = a(n-1) + a(n-2) + A006054(n+1). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 05 2008

A006356:=-(-1-z+z**2)/(1-2*z-z**2+z**3); [Conjectured by S. Plouffe in his 1992 dissertation.]

(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)

Cf. A000217, A000330, A050446, A050447, A006054, A077998, A052534, A052994, A052949.

See also A006357-A006359, A025030, A030112-A030116.

Cf. A038196 (3-wave sequence).

Adjacent sequences: A006353 A006354 A006355 this_sequence A006357 A006358 A006359

Sequence in context: A091601 A063119 A106803 this_sequence A077998 A090165 A129954

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

Recurrence, alternative description from Jacques Haubrich (jhaubrich(AT)freeler.nl).

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Dec 24 1999

Added an alternative definition. - Andy Niedermaier (aniederm(AT)math.ucsd.edu), Nov 11 2008