A091024 -id:A091024 - cp's OEIS frontend

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

1, 4, 10, 30, 85, 246, 707, 2037, 5864, 16886, 48620, 139997, 403104, 1160693, 3342081, 9623140, 27708726, 79784098, 229729153, 661478734, 1904652103, 5484227157, 15791202736, 45468956106, 130922641160, 376976720745, 1085461206128, 3125460977225

Offset: 0

N. J. A. Sloane

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
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
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

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, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 120).
J. Haubrich, Multinacci Rijen [Multinacci sequences], Euclides (Netherlands), Vol. 74, Issue 4, 1998, pp. 131-133.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..200
J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]
Emma L. L. Gao, Sergey Kitaev, and Philip B. Zhang, Pattern-avoiding alternating words, arXiv:1505.04078 [math.CO], 2015.
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.
G. Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30.
G. Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30. (Annotated scanned copy)
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (2,3,-1,-1).

Cf. A000217, A000330, A050446, A050447, A006356-A006359, A025030, A030112-A030116, A122167, A091024.

Cf. A038197 (4-wave sequence).

LinearRecurrence[{2,3,-1,-1},{1,4,10,30},30] (* Harvey P. Dale, Nov 18 2013 *)

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

G.f.: (1 + 2*x - x^2 - x^3)/( (1 +x)*(1 -3*x +x^3) ). - Simon Plouffe in his 1992 dissertation
a(n) = 2*a(n-1) + 3*a(n-2) - a(n-3) - a(n-4).
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
Binomial transform of A122167(unsigned): (1, 3, 3, 11, 10, 40, 33, 146, ...). - Gary W. Adamson, Nov 24 2007
G.f.: 1/(-x-1/(-x-1/(-x-1/(-x-1)))). - Paul Barry, Mar 24 2010

Recurrence, alternative description from Jacques Haubrich (jhaubrich(AT)freeler.nl)
More terms from James Sellers, Dec 24 1999
More terms from Paul D. Hanna, Feb 06 2006

A038197 4-wave sequence.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 7, 9, 10, 19, 26, 30, 56, 75, 85, 160, 216, 246, 462, 622, 707, 1329, 1791, 2037, 3828, 5157, 5864, 11021, 14849, 16886, 31735, 42756, 48620, 91376, 123111, 139997, 263108, 354484, 403104, 757588, 1020696, 1160693, 2181389

Offset: 0

Floor van Lamoen

This sequence is related to the nonagon or 9-gon.

			The first few rows of the T(n,k) array are, n>=1, 1 <= k <=4:
  0,  0,   0,   1
  1,  1,   1,   1
  1,  2,   3,   4
  4,  7,   9,   10
  10, 19,  26,  30
  30, 56,  75,  85
  85, 160, 216, 246

F. v. Lamoen, Wave sequences
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
Eric Weisstein's World of Mathematics, Nonagon.
Index entries for linear recurrences with constant coefficients, signature (1,-1,3,-3,3,0,0,0,-1,1,-1).

The a(3*n) lead to A006357; The T(n,k) lead to A076264 and A091024.
Cf. A038196, A038201.
Cf. A120747 (m = 5: hendecagon or 11-gon)

m:=4: nmax:=15: for k from 1 to m-1 do T(1,k):=0 od: T(1,m):=1: for n from 2 to nmax do for k from 1 to m do T(n,k):= add(T(n-1,k1), k1=m-k+1..m) od: od: for n from 1 to nmax/2 do seq(T(n,k), k=1..m) od; a(0):=1: Tx:=1: for n from 2 to nmax do for k from 2 to m do a(Tx):= T(n,k): Tx:=Tx+1: od: od: seq(a(n), n=0..Tx-1); # Johannes W. Meijer, Aug 03 2011

LinearRecurrence[{1,-1,3,-3,3,0,0,0,-1,1,-1},{1,1,1,1,2,3,4,7,9,10,19},50] (* Harvey P. Dale, Oct 02 2015 *)

a(n) = a(n-1)+a(n-2) if n=3*m+1, a(n) = a(n-1)+a(n-4) if n=3*m+2, a(n) = a(n-1)+a(n-6) if n=3*m. Also: a(n) = 2*a(n-3)+3*a(n-6)-a(n-9)-a(n-12).
G.f.: -(-1-x-x^2+x^3-x^5+x^6)/(1-2*x^3-3*x^6+x^9+x^12)
a(n-1) = sequence(sequence(T(n,k), k=2..4), n>=2) with a(0)=1; T(n,k) = sum(T(n-1,k1), k1 = 5-k..4) with T(1,1) = T(1,2) = T(1,3) = 0 and T(1,4) = 1; n>=1 and 1 <= k <= 4. [Steinbach]

Edited by Floor van Lamoen, Feb 05 2002

Edited and information added by Johannes W. Meijer, Aug 03 2011

A120747 Sequence relating to the 11-gon (or hendecagon).

Original entry on oeis.org

0, 1, 4, 14, 50, 175, 616, 2163, 7601, 26703, 93819, 329615, 1158052, 4068623, 14294449, 50221212, 176444054, 619907431, 2177943781, 7651850657, 26883530748, 94450905714, 331837870408, 1165858298498, 4096053203771, 14390815650209, 50559786403254

Offset: 1

Gary W. Adamson, Jul 01 2006

The hendecagon is an 11-sided polygon. The preferred word in the OEIS is 11-gon.
The lengths of the diagonals of the regular 11-gon are r[k] = sin(k*Pi/11)/sin(Pi/11), 1 <= k <= 5, where r[1] = 1 is the length of the edge.
The value of limit(a(n)/a(n-1),n=infinity) equals the longest diagonal r[5].
The a(n) equal the matrix elements M^n[1,2], where M = Matrix([[1,1,1,1,1], [1,1,1,1,0], [1,1,1,0,0], [1,1,0,0,0], [1,0,0,0,0]]). The characteristic polynomial of M is (x^5 - 3x^4 - 3x^3 + 4x^2 + x - 1) with roots x1 = -r[4]/r[3], x2 = -r[2]/r[4], x3 = r[1]/r[2], x4 = r[3]/r[5] and x5 = r[5]/r[1].
Note that M^4*[1,0,0,0,0] = [55, 50, 41, 29, 15] which are all terms of the 5-wave sequence A038201. This is also the case for the terms of M^n*[1,0,0,0,0], n>=1.

			From _Johannes W. Meijer_, Aug 03 2011: (Start)
The lengths of the regular hendecagon edge and diagonals are:
  r[1] = 1.000000000, r[2] = 1.918985948, r[3] = 2.682507066,
  r[4] = 3.228707416, r[5] = 3.513337092.
The first few rows of the T(n,k) array are, n>=1, 1 <= k <=5:
    0,   0,   0,   0,   1, ...
    1,   1,   1,   1,   1, ...
    1,   2,   3,   4,   5, ...
    5,   9,  12,  14,  15, ...
   15,  29,  41,  50,  55, ...
   55, 105, 146, 175, 190, ...
  190, 365, 511, 616, 671, ... (End)

G. C. Greubel, Table of n, a(n) for n = 1..1000
Jay Kappraff, Slavik Jablan, Gary W. Adamson and Radmila Sazdanovich, Golden Fields, Generalized Fibonacci Sequences and Chaotic Matrices, Forma, Vol. 19 No. 4, pp. 367-387, 2004.
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31, MR 1439165
Eric Weisstein's World of Mathematics, Hendecagon.
Index entries for linear recurrences with constant coefficients, signature (3,3,-4,-1,1).

Cf. A006358, A033304, A038201, A052963, A065941, A066170.
From Johannes W. Meijer, Aug 03 2011: (Start)
Cf. A006358 (T(n+2,1) and T(n+1,5)), A069006 (T(n+1,2)), A038342 (T(n+1,3)), this sequence (T(n,4)) (m=5: hendecagon or 11-gon).
Cf. A000045 (m=2; pentagon or 5-gon); A006356, A006054 and A038196 (m=3: heptagon or 7-gon); A006357, A076264, A091024 and A038197 (m=4: enneagon or 9-gon); A006359, A069007, A069008, A069009, A070778 (m=6; tridecagon or 13-gon); A025030 (m=7: pentadecagon or 15-gon); A030112 (m=8: heptadecagon or 17-gon). (End)

R:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x^2*(1+x-x^2)/(1-3*x-3*x^2+4*x^3+x^4-x^5) )); // G. C. Greubel, Nov 13 2022

nmax:=27: m:=5: for k from 1 to m-1 do T(1,k):=0 od: T(1,m):=1: for n from 2 to nmax do for k from 1 to m do T(n,k):= add(T(n-1,k1), k1=m-k+1..m) od: od: for n from 1 to nmax/3 do seq(T(n,k), k=1..m) od; for n from 1 to nmax do a(n):=T(n,4) od: seq(a(n), n=1..nmax); # Johannes W. Meijer, Aug 03 2011

LinearRecurrence[{3, 3, -4, -1, 1}, {0, 1, 4, 14, 50}, 41] (* G. C. Greubel, Nov 13 2022 *)

def A120747_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1+x-x^2)/(1-3*x-3*x^2+4*x^3+x^4-x^5) ).list()
A120747_list(40) # G. C. Greubel, Nov 13 2022

a(n) = 3*a(n-1) + 3*a(n-2) - 4*a(n-3) - a(n-4) + a(n-5).
G.f.: x^2*(1+x-x^2)/(1-3*x-3*x^2+4*x^3+x^4-x^5). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 12 2009
From Johannes W. Meijer, Aug 03 2011: (Start)
a(n) = T(n,4) with T(n,k) = Sum_{k1 = 6-k..6} T(n-1, k1), T(1,1) = T(1,2) = T(1,3) = T(1,4) = 0 and T(1,5) = 1, n>=1 and 1 <= k <= 5. [Steinbach]
Sum_{k=1..5} T(n,k)*r[k] = r[5]^n, n>=1. [Steinbach]
r[k] = sin(k*Pi/11)/sin(Pi/11), 1 <= k <= 5. [Kappraff]
Sum_{k=1..5} T(n,k) = A006358(n-1).
Limit_{n -> 00} T(n,k)/T(n-1,k) = r[5], 1 <= k <= 5.
sequence(sequence( T(n,k), k=2..5), n>=1) = A038201(n-4).
G.f.: (x^2*(x - x1)*(x - x2))/((x - x3)*(x - x4)*(x - x5)*(x - x6)*(x - x7)) with x1 = phi, x2 = (1-phi), x3 = r[1] - r[3], x4 = r[3] - r[5], x5 = r[5] - r[4], x6 = r[4] - r[2], x7 = r[2], where phi = (1 + sqrt(5))/2 is the golden ratio A001622. (End)

Edited and information added by Johannes W. Meijer, Aug 03 2011

A123609 Quasiperiodic 9-gonal (nonagonal) sequence as a 1-dimensional tiling.

Original entry on oeis.org

4, 1, 2, 3, 4, 4, 3, 4, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 2, 3, 4, 1, 2, 3, 4, 3, 4, 2, 3, 4, 1, 2, 3, 4, 4, 3, 4, 2, 3, 4, 1, 2, 3, 4

Offset: 1

Gary W. Adamson and Roger L. Bagula, Oct 03 2006

The frequency of each distinct term (1, 2, 3, or 4) tends to converge to the ratio of each diagonal (a, b, c, or d) to the sum of the 4 diagonal lengths. The four 9-gon (nonagon) diagonals are a=1, b=1.87938524..., c=2.53208888... and d=2.87938524..., with the sum a+b+c+d = d^2 = 8.29085936.... Converting these terms to fractions of the sum, a=0.12061475..., b=0.22668159..., c=0.30540728..., and d=0.34729635.... Through n = 45, we can thus expect sixteen 4's (correct), since round(45*.34729635...) = 16. The numbers of terms in each subset strung together is found in A006357: (1, 4, 10, 30, 85...), thus: (4), (1,2,3,4), (4,3,4,2,3,4,1,2,3,4), ..., while the distributive breakdown of numbers of 1's, 2's, 3's, and 4's may be found in the 4-termed set of vectors in A076264: 1 1 1 1 4 3 2 1 10 9 7 4 30 26 19 10 ... where the sum of 4 terms in a row = the left term in the next row. For example, the frequency distribution of 30 includes ten 4's, nine 3's, seven 2's, and four 1's. Check: the subset of 30 terms generated from the previous subset of 10: (1,2,3,4,2,3,4,1,2,3,4,3,4,2,3,4,1,2,3,4,4,3,4,2,3,4,1,2,3,4).

A fractal structure is suggested by parsing each subset into groups: (1,2,3,4), (2,3,4), (1,2,3,4), (3,4), (2,3,4), (1,2,3,4), (4), (3,4), (2,3,4), (1,2,3,4). That is, 10 groups: four with four terms, three with three terms, two with two terms, and one with one term. Replacing the terms (4,3,2,1) with the diagonal lengths (d,c,b,a) and referring to the set of vectors: (1,1,1,1; 4,3,2,1; 10,9,7,4; ...), label these rows 2,3,4,... and consider (2,3,4,...) exponents to diagonal d=2.87938524..., such that, for example, "4" corresponds to (10,9,7,4), and (Cf. Steinbach) d^4 = 68.738349... = (10*d + 9*c + 7*b + 4*a). Such relationships are a consequence of the "Diagonal Product Formulas" mentioned on p. 23.

			1=>4, then 4=>1,2,3,4, which, in turn, generates 4,3,4,2,3,4,1,2,3,4 (append next result to right of previous result, getting an infinite aperiodic sequence).

P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31, p. 29.

Cf. A123508, A006357, A076264, A091024.

Drop[SubstitutionSystem[{1->{4},2->{3,4},3->{2,3,4},4->{1,2,3,4}},{1},{5}][[1]],5] (* Harvey P. Dale, Mar 02 2022 *)

Using the seed "1", we use the recurrence rules 1=>4; 2=>3,4; 3=>2,3,4; 4=>1,2,3,4; to form iterative subsets which are appended in succession to form a continuous string.

Partially edited by Jon E. Schoenfield, Sep 15 2013

A069005 Let M = 4 X 4 matrix with rows /1,1,1,1/1,1,1,0/1,1,0,0/1,0,0,0/ and A(n) = vector (x(n),y(n),z(n),t(n)) = M^n*A where A is the vector (1,1,1,1); then a(n)=z(n).

Original entry on oeis.org

1, 7, 19, 56, 160, 462, 1329, 3828, 11021, 31735, 91376, 263108, 757588, 2181389, 6281058, 18085587, 52075371, 149945056, 431749580, 1243173370, 3579575053, 10306975580, 29677753369, 85453685055, 246054079584, 708484485384

Offset: 1

Benoit Cloitre, Apr 02 2002

a(n) = A091024(n+1) for n > 1. - Georg Fischer, Oct 19 2018

Index entries for linear recurrences with constant coefficients, signature (2,3,-1,-1).

Cf. A018919, A091024.

CoefficientList[Series[x (-x^4 - 2 x^3 + 2 x^2 + 5 x + 1)/((1 + x) (1 - 3 x + x^3)), {x, 0, 40}], x] (* Georg Fischer, May 24 2019 *)

G.f.: x*(-x^4-2*x^3+2*x^2+5*x+1)/((1+x)*(1-3*x+x^3)). [Corrected by Georg Fischer, May 24 2019]

cp's OEIS Frontend

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

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A038197 4-wave sequence.

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A120747 Sequence relating to the 11-gon (or hendecagon).

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A123609 Quasiperiodic 9-gonal (nonagonal) sequence as a 1-dimensional tiling.

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A069005 Let M = 4 X 4 matrix with rows /1,1,1,1/1,1,1,0/1,1,0,0/1,0,0,0/ and A(n) = vector (x(n),y(n),z(n),t(n)) = M^n*A where A is the vector (1,1,1,1); then a(n)=z(n).

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