A006356 -id:A006356 - cp's OEIS frontend

A056932 Antichains (or order ideals) in the poset 222n or size of the distributive lattice J(222n).

1, 20, 168, 887, 3490, 11196, 30900, 75966, 170379, 354640, 693836, 1288365, 2287844, 3908776, 6456600, 10352796, 16167765, 24660252, 36824128, 53943395, 77656326, 110029700, 153644140, 211691610, 288086175, 387589176, 515950020, 680063833, 888147272

Offset: 0

Mitch Harris

a(n) is the number of order preserving maps from B_3 into [n+1]. a(n) is also the number of length n+1 multichains from bottom to top in J(B_3). See Stanley reference for bijections with description in title. - Geoffrey Critzer, Jan 07 2021

J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124.
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. P. Stanley, Enumerative Combinatorics, Volume I, Second Edition, page 256, Proposition 3.5.1.

T. D. Noe, Table of n, a(n) for n = 0..1000
J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]
G. Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 9.
Index entries for sequences related to posets
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A000372, A006360, A006361, A006362, A056933, A056934, A056935, A056936, A056937.

Table[48*Binomial[n+8,8] - 96*Binomial[n+7,7] + 63*Binomial[n+6,6] - 15*Binomial[n+5,5] + Binomial[n+4,4], {n, 0, nn}] (* T. D. Noe, May 29 2012 *)

a(n) = 48*C(n+8, 8) - 96*C(n+7, 7) + 63*C(n+6, 6) - 15*C(n+5, 5) + C(n+4, 4).

G.f.: (1+11*x+24*x^2+11*x^3+x^4)/(1-x)^9. [Berman and Koehler]

A212232 T(n,k)=Number of 0..2 arrays of length n+2k-1 with sum no more than 2k in any length 2k subsequence (=50% duty cycle).

Original entry on oeis.org

6, 50, 14, 435, 124, 31, 3834, 1113, 311, 70, 34001, 10002, 2902, 775, 157, 302615, 89911, 26637, 7596, 1895, 353, 2699598, 808403, 242780, 71427, 19834, 4663, 793, 24121674, 7269626, 2204646, 660796, 191853, 51440, 11518, 1782, 215786649

Offset: 1

R. H. Hardin May 06 2012

Table starts
....6....50....435....3834....34001....302615....2699598....24121674
...14...124...1113...10002....89911....808403....7269626....65380788
...31...311...2902...26637...242780...2204646...19976155...180744711
...70...775...7596...71427...660796...6062948...55360211...503916387
..157..1895..19834..191853..1804448..16740414..154089343..1411275807
..353..4663..51440..514687..4931630..46305966..429886243..3962696059
..793.11518.131950.1376128.13468524.128148456.1200645159.11143104246
.1782.28446.339564.3659968.36711516.354470546.3354267511.31356940932

			Some solutions for n=3 k=4
..1....2....2....1....2....0....1....2....2....1....1....2....0....0....1....1
..1....0....0....1....1....1....0....2....2....0....0....2....0....1....1....1
..1....2....2....0....0....0....0....1....0....1....0....0....1....0....0....2
..0....0....1....1....1....2....2....1....1....1....0....0....1....0....1....1
..0....1....0....0....1....2....0....1....1....1....0....1....0....1....1....2
..0....0....2....1....1....0....1....1....1....1....1....2....1....0....0....0
..1....1....0....1....1....0....0....0....0....2....2....1....0....2....1....0
..1....1....0....1....0....0....0....0....0....0....2....0....1....1....1....0
..1....1....1....1....1....1....0....1....1....1....2....0....0....1....0....0
..0....1....2....2....2....0....0....2....1....1....1....1....2....1....0....1

R. H. Hardin, Table of n, a(n) for n = 1..1074

Column 1 is A006356(n+1)

A213464 T(n,k)=Number of 0..3 arrays of length n+2k-1 with sum less than 3k in any length 2k subsequence (=less than 50% duty cycle).

Original entry on oeis.org

6, 106, 14, 1758, 310, 31, 28722, 5542, 927, 70, 466136, 94238, 18018, 2735, 157, 7536790, 1568984, 319078, 58898, 7803, 353, 121573668, 25829770, 5444644, 1091036, 191573, 22506, 793, 1957953138, 422335348, 91147915, 19105020, 3737885

Offset: 1

R. H. Hardin Jun 12 2012

Table starts
....6....106....1758.....28722.....466136.....7536790....121573668
...14....310....5542.....94238....1568984....25829770....422335348
...31....927...18018....319078....5444644....91147915...1508740928
...70...2735...58898...1091036...19105020...325333429...5451265872
..157...7803..191573...3737885...67329406..1167362677..19808276303
..353..22506..615561..12773955..237475982..4198162231..72188618371
..793..65425.1940673..43399990..836356119.15103463125.263422708979
.1782.190318.6155514.146145812.2935825527.54284048195.961420842175

			Some solutions for n=3 k=4
..2....0....2....0....0....0....2....0....0....2....0....0....0....2....0....0
..2....2....0....2....1....2....2....0....0....0....0....1....1....2....2....0
..2....1....0....0....0....3....0....0....0....0....0....2....0....1....1....0
..1....3....1....1....0....0....0....2....0....0....2....0....0....1....1....1
..1....1....1....3....2....0....2....2....0....3....3....0....0....1....0....2
..0....1....3....1....2....2....1....2....0....0....0....0....0....2....2....0
..1....0....0....0....1....0....1....0....0....0....1....0....2....1....1....1
..1....1....0....0....1....0....3....0....0....2....0....2....2....0....2....2
..2....0....2....1....3....0....2....1....2....2....1....0....2....2....2....2
..3....3....0....3....0....2....1....3....2....3....2....0....2....1....1....2

R. H. Hardin, Table of n, a(n) for n = 1..897

Column 1 is A006356(n+1)

A006358 Number of distributive lattices; also number of paths with n turns when light is reflected from 5 glass plates.

Original entry on oeis.org

1, 5, 15, 55, 190, 671, 2353, 8272, 29056, 102091, 358671, 1260143, 4427294, 15554592, 54648506, 191998646, 674555937, 2369942427, 8326406594, 29253473175, 102777312308, 361091343583, 1268635610806, 4457144547354

Offset: 0

N. J. A. Sloane

Let M denotes the 5 X 5 matrix = row by row (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) and A(n) the vector (x(n),y(n),z(n),t(n),u(n)) = M^n*A where A is the vector (1,1,1,1,1); then a(n)=y(n). - Benoit Cloitre, Apr 02 2002

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.
D. E. Knuth, Art of Computer Programming, Vol. 3, Sect. 5.4.3, Column T1.
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 (3,3,-4,-1,1).

Cf. A000217, A000330, A050446, A050447.
See also A006356-A006359, A025030, A030112-A030116.
Cf. A038201 (5-wave sequence).

A=seq(a.j,j=0..4):grammar1:=[Q4,{ seq(Q.i=Union(Epsilon,seq(Prod(a.j,Q.j),j=4-i..4)),i=0..4), seq(a.j=Z,j=0..4) }, unlabeled]: seq(count(grammar1,size=j),j=0..23); # Zerinvary Lajos, Mar 09 2007
A006358:=-(z-1)*(z**3-3*z-1)/(-1+3*z+3*z**2-4*z**3-z**4+z**5); # conjectured by Simon Plouffe in his 1992 dissertation

m = Table[ If[j <= 6-i, 1, 0], {i, 1, 5}, {j, 1, 5}] ; a[n_] := MatrixPower[m, n].Table[1, {5}]; Table[ a[n], {n, 0, 23}][[All, 1]] (* Jean-François Alcover, Dec 08 2011, after Benoit Cloitre *)
LinearRecurrence[{3,3,-4,-1,1},{1,5,15,55,190},30] (* Harvey P. Dale, Jun 16 2016 *)

k=5; M(k)=matrix(k,k,i,j,if(1-sign(i+j-k),0,1)); v(k)=vector(k,i,1); a(n)=vecmax(v(k)*M(k)^n)

{a(n)=local(p=5);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)}

a(n) = 3*a(n-1) + 3*a(n-2) - 4*a(n-3) - a(n-4) + a(n-5).
a(n) is asymptotic to z(5)*w(5)^n where w(5) = (1/2)/cos(5*Pi/11) and z(5) is the root 1 < x < 2 of P(5, X) = -1 + 55*X + 847*X^2 - 5324*X^3 - 14641*X^4 + 14641*X^5. - Benoit Cloitre, Oct 16 2002
G.f.: A(x) = (1 + 2*x - 3*x^2 - x^3 + x^4)/(1 - 3*x - 3*x^2 + 4*x^3 + x^4 - x^5). - Paul D. Hanna, Feb 06 2006

Alternative description and formula from Jacques Haubrich (jhaubrich(AT)freeler.nl)

More terms from James Sellers, Dec 24 1999

A038196 3-wave sequence starting with 1, 1, 1.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 6, 11, 14, 25, 31, 56, 70, 126, 157, 283, 353, 636, 793, 1429, 1782, 3211, 4004, 7215, 8997, 16212, 20216, 36428, 45425, 81853, 102069, 183922, 229347, 413269, 515338, 928607, 1157954, 2086561, 2601899, 4688460, 5846414

Offset: 0

Floor van Lamoen

The 3-wave sequence with initial values a, b, c is formed by the following construction:
a.......a+b+c............3a+5b+6c...
..b...b+c...a+2b+2c..2a+4b+5c...
....c..........a+2b+3c...

J. Kappraff, Beyond Measure, World Scientific, Inc. 2002, p. 497.

F. v. Lamoen, Wave sequences
Index entries for linear recurrences with constant coefficients, signature (0, 2, 0, 1, 0, -1).

a(2n) forms A006356, a(2n+1) ("the middle row") forms A006054. Cf. A038197, A038201, A187070.

a(n)=if(n>-1,polcoeff((1+x-x^2)/(1-2*x^2-x^4+x^6)+x*O(x^n),n),if(n<-3,polcoeff((1-x-x^2)/(1-x^2-2*x^4+x^6)+O(x^(-3-n)),-4-n),0))

a(n) = a(n-1) + a(n-2) if n is odd,
a(n) = a(n-1) + a(n-4) if n is even.
Also: a(n) = 2*a(n-2) + a(n-4) - a(n-6).
G.f.: (1 + x - x^2)/(1 - 2*x^2 - x^4 + x^6).

Edited by Floor van Lamoen, Feb 05 2002

A025030 Number of distributive lattices; also number of paths with n turns when light is reflected from 7 glass plates.

Original entry on oeis.org

1, 7, 28, 140, 658, 3164, 15106, 72302, 345775, 1654092, 7911970, 37846314, 181033035, 865951710, 4142180085, 19813648817, 94776329265, 453351783116, 2168556616440, 10373043626906, 49618272850056, 237343357526002

Offset: 0

Jacques Haubrich (jhaubrich(AT)freeler.nl)

Let M(7) be the 7 X 7 matrix: (0,0,0,0,0,0,1)/(0,0,0,0,0,1,1)/(0,0,0,0,1,1,1)/(0,0,0,1,1,1,1)/(0,0,1,1,1,1,1)/(0,1,1,1,1,1,1)/(1,1,1,1,1,1,1) and let v(7) be the vector (1,1,1,1,1,1,1); then v(7)*M(7)^n = (x,y,z,t,u,v,a(n)). - Benoit Cloitre, Sep 29 2002

J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124.
J. Haubrich, Multinacci Rijen [Multinacci sequences], Euclides (Netherlands), Vol. 74, Issue 4, 1998, pp. 131-133.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]
G. Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30.
Index entries for linear recurrences with constant coefficients, signature (4,6,-10,-5,6,1,-1).

See also A006356-A006359, A030112-A030116.

I:=[1, 7, 28, 140, 658, 3164, 15106]; [n le 7 select I[n] else 4*Self(n-1)+6*Self(n-2)-10*Self(n-3)-5*Self(n-4)+6*Self(n-5)+Self(n-6)-Self(n-7): n in [1..30]]; // Vincenzo Librandi, Apr 22 2012

CoefficientList[Series[(1+3*x-6*x^2-4*x^3+5*x^4+x^5-x^6)/((1-x)*(1+x-x^2)*(1-4*x-4*x^2+x^3+x^4)),{x,0,30}],x] (* Vincenzo Librandi, Apr 22 2012 *)
LinearRecurrence[{4,6,-10,-5,6,1,-1},{1,7,28,140,658,3164,15106},30] (* Harvey P. Dale, Feb 26 2023 *)

k=7; M(k)=matrix(k,k,i,j,if(1-sign(i+j-k),0,1)); v(k)=vector(k,i,1); a(n)=vecmax(v(k)*M(k)^n)

a(n) = 4*a(n-1) + 6*a(n-2) - 10*a(n-3) - 5*a(n-4) + 6*a(n-5) + a(n-6) - a(n-7).
a(n) is asymptotic to z(7)*w(7)^n where w(7) = (1/2)/cos(7*Pi/15) and z(7) is the root 1 < x < 2 of P(7, X) = 1 - 120*X - 8100*X^2 - 57375*X^3 + 50625*X^4. - Benoit Cloitre, Oct 16 2002
G.f.: (1 + 3*x - 6*x^2 - 4*x^3 + 5*x^4 + x^5 - x^6)/((1 - x)*(1 + x - x^2)*(1 - 4*x - 4*x^2 + x^3 + x^4)). - Colin Barker, Mar 31 2012

More terms from Benoit Cloitre, Sep 29 2002

A030112 Number of distributive lattices; also number of paths with n turns when light is reflected from 8 glass plates.

Original entry on oeis.org

1, 8, 36, 204, 1086, 5916, 31998, 173502, 940005, 5094220, 27604798, 149590922, 810627389, 4392774126, 23804329059, 128995094597, 699021261776, 3787979292364, 20526967746120, 111235140046330, 602780523265720, 3266453022809170, 17700829632401740, 95920366069513405

Offset: 0

Jacques Haubrich (jhaubrich(AT)freeler.nl)

Let M(8) be the 8 X 8 matrix (0,0,0,0,0,0,0,1)/(0,0,0,0,0,0,1,1)/(0,0,0,0,0,1,1,1)/(0,0,0,0,1,1,1,1)/(0,0,0,1,1,1,1,1)/(0,0,1,1,1,1,1,1)/(0,1,1,1,1,1,1,1)/(1,1,1,1,1,1,1,1) and let v(8) be the vector (1,1,1,1,1,1,1,1); then v(8)*M(8)^n = (x,y,z,t,u,v, w,a(n)). - Benoit Cloitre, Sep 29 2002

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 1Benoit Cloitre, Oct 16 2002

J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124.
J. Haubrich, Multinacci Rijen [Multinacci sequences], Euclides (Netherlands), Vol. 74, Issue 4, 1998, pp. 131-133.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]
G. Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30.
Index entries for linear recurrences with constant coefficients, signature (4,10,-10,-15,6,7,-1,-1).

See also A006356-A006359, A025030, A030113-A030116.

I:=[1, 8, 36, 204, 1086, 5916, 31998, 173502]; [n le 8 select I[n] else 4*Self(n-1)+10*Self(n-2)-10*Self(n-3)-15*Self(n-4)+6*Self(n-5)+7*Self(n-6)-Self(n-7)-Self(n-8):  n in [1..25]]; // Vincenzo Librandi, Apr 22 2012

nmax:=20: with(LinearAlgebra): M:=Matrix([[0,0,0,0,0,0,0,1], [0,0,0,0,0,0,1,1], [0,0,0,0,0,1,1,1], [0,0,0,0,1,1,1,1], [0,0,0,1,1,1,1,1], [0,0,1,1,1,1,1,1], [0,1,1,1,1,1,1,1], [1,1,1,1,1,1,1,1]]): v:= Vector[row]([1,1,1,1,1,1,1,1]): for n from 0 to nmax do b:=evalm(v&*M^n): a(n):=b[8] od: seq(a(n), n=0..nmax); # Johannes W. Meijer, Aug 03 2011

CoefficientList[Series[(1+x)*(1-x-x^2)*(1+4*x-4*x^2-x^3+x^4)/(1-4*x-10*x^2+10*x^3+15*x^4-6*x^5-7*x^6+x^7+x^8),{x,0,30}],x] (* Vincenzo Librandi, Apr 22 2012 *)

k=8; M(k)=matrix(k,k,i,j,if(1-sign(i+j-k),0,1)); v(k)=vector(k,i,1); a(n)=vecmax(v(k)*M(k)^n)

a(n) = 4*a(n-1)+ 10*a(n-2)-10*a(n-3)-15*a(n-4)+ 6*a(n-5)+7*a(n-6)-a(n-7)-a(n-8). - Benoit Cloitre, Oct 09 2002
a(n) is asymptotic to z(8)*w(8)^n where w(8)=(1/2)/cos(8*Pi/17) and z(8) is the root 1Benoit Cloitre, Oct 16 2002
G.f.: (1+x)*(1-x-x^2)*(1+4*x-4*x^2-x^3+x^4)/(1-4*x-10*x^2+10*x^3+15*x^4-6*x^5-7*x^6+x^7+x^8). - Colin Barker, Mar 31 2012

More terms from Benoit Cloitre, Sep 29 2002

Comment corrected by Johannes W. Meijer, Aug 03 2011

A052534 Expansion of (1-x)(1+x)/(1-2x-x^2+x^3).

Original entry on oeis.org

1, 2, 4, 9, 20, 45, 101, 227, 510, 1146, 2575, 5786, 13001, 29213, 65641, 147494, 331416, 744685, 1673292, 3759853, 8448313, 18983187, 42654834, 95844542, 215360731, 483911170, 1087338529, 2443227497, 5489882353, 12335653674

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Pairwise sums of A006356. Cf. A033303, A077850. - Ralf Stephan, Jul 06 2003
Number of (3412, P)-avoiding involutions in S_{n+1}, where P={1342, 1423, 2314, 3142, 2431, 4132, 3241, 4213, 21543, 32154, 43215, 15432, 53241, 52431, 42315, 15342, 54321}. - Ralf Stephan, Jul 06 2003
Number of 31- and 22-avoiding words of length n on alphabet {1,2,3} which do not end in 3 (e.g., at n=3, we have 111, 112, 121, 132, 211, 212, 232, 321 and 332). See A028859, A001519. - Jon Perry, Aug 04 2003
Form the graph with matrix A=[1, 1, 1; 1, 0, 0; 1, 0, 1]. Then the sequence 1,1,2,4,... with g.f. (1-x-x^2)/(1-2x-x^2+x^3) counts closed walks of length n at the degree 3 vertex. - Paul Barry, Oct 02 2004
a(n) is the number of Motzkin (n+1)-sequences whose flatsteps all occur at level <=1 and whose height is <=2. For example, a(5)=45 counts all 51 Motzkin 6-paths except FUUFDD, UFUFDD, UUFDDF, UUFDFD, UUFFDD, UUUDDD (the first five violate the flatstep restriction and the last violates the height restriction). - David Callan, Dec 09 2004
From Paul Barry, Nov 03 2010: (Start)
The g.f. of 1,1,2,4,9,... can be expressed as 1/(1-x/(1-x/(1-x^2))) and as 1/(1-x-x^2/(1-x-x^2)).
The second expression shows the link to the Motzkin numbers. (End)
From Emeric Deutsch, Oct 31 2010: (Start)
a(n) is the number of compositions of n into odd summands when we have two kinds of 1's. Proof: the g.f. of the set S={1,1',3,5,7,...} is g=2x+x^3/(1-x^2) and the g.f. of finite sequences of elements of S is 1/(1-g). Example: a(4)=20 because we have 1+3, 1'+3, 3+1, 3+1', and 2^4=16 of sums x+y+z+u, where x,y,z,u are taken from {1,1'}.
(End)
a(n-1) is the top left entry of the n-th power of any of the six 3 X 3 matrices [1, 1, 0; 1, 1, 1; 0, 1, 0] or [1, 1, 1; 0, 1, 1; 1, 1, 0] or [1, 0, 1; 1, 1, 1; 1, 1, 0] or [1, 1, 1; 1, 0, 1; 0, 1, 1] or [1, 0, 1; 0, 0, 1; 1, 1, 1] or [1, 1, 0; 1, 0, 1; 1, 1, 1]. - R. J. Mathar, Feb 03 2014

			G.f. = 1 + 2*x + 4*x^2 + 9*x^3 + 20*x^4 + 45*x^5 + 101*x^6 + 227*x^7 + 510*x^8 + ... - _Michael Somos_, Dec 12 2023

G. C. Greubel, Table of n, a(n) for n = 0..1000
Jean-Luc Baril, Rigoberto Flórez, and José L. Ramírez, Counting symmetric and asymmetric peaks in motzkin paths with air pockets, Univ. Bourgogne (France, 2023).
Nicolas Bělohoubek and Antonín Slavík, L-Tetromino Tilings and Two-Color Integer Compositions, Univ. Karlova (Czechia, 2025). See p. 10.
C. P. de Andrade, J. P. de Oliveira Santos, E. V. P. da Silva and K. C. P. Silva, Polynomial Generalizations and Combinatorial Interpretations for Sequences Including the Fibonacci and Pell Numbers, Open Journal of Discrete Mathematics, 2013, 3, 25-32 doi:10.4236/ojdm.2013.31006. - From _N. J. A. Sloane_, Feb 20 2013
E. S. Egge, Restricted 3412-Avoiding Involutions: Continued Fractions, Chebyshev Polynomials and Enumerations, arXiv:math/0307050 [math.CO], 2003, sec. 8.
Rigoberto Flórez and José L. Ramírez, Some enumerations on non-decreasing Motzkin paths, Australasian J. of Combinatorics (2018) Vol. 72(1), 138-154.
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See p. 15.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 464
S. Morier-Genoud, V. Ovsienko, and S. Tabachnikov, Introducing supersymmetric frieze patterns and linear difference operators, Math. Z. 281 (2015) 1061.
Alexey Ustinov, Supercontinuants, arXiv:1503.04497 [math.NT], 2015.
R. Witula, D. Slota and A. Warzynski, Quasi-Fibonacci Numbers of the Seventh Order, J. Integer Seq., 9 (2006), Article 06.4.3.
Index entries for linear recurrences with constant coefficients, signature (2,1,-1).

Cf. A028495, A066170.

a:=[1,2,4];; for n in [4..40] do a[n]:=2*a[n-1]+a[n-2]-a[n-3]; od; a; # G. C. Greubel, May 09 2019

[n le 3 select 2^(n-1) else 2*Self(n-1)+Self(n-2)-Self(n-3): n in [1..40]]; // Vincenzo Librandi, Mar 17 2015

spec := [S,{S=Sequence(Union(Z,Prod(Z,Sequence(Prod(Z,Z)))))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

LinearRecurrence[{2,1,-1},{1,2,4},40] (* Roman Witula, Aug 07 2012 *)
CoefficientList[Series[(1-x^2)/(1-2x-x^2+x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 17 2015 *)
a[ n_] := {0, 1, 0} . MatrixPower[{{1, 1, 1}, {1, 1, 0}, {1, 0, 0}}, n+1] . {0, 1, 0}; (* Michael Somos, Dec 12 2023 *)

h(n):=if n=0 then 1 else sum(sum(binomial(k,j)*binomial(j,n-3*k+2*j)*2^(3*k-n-j)*(-1)^(k-j),j,0,k),k,1,n); a(n):=if n<2 then h(n) else h(n)-h(n-2); /* Vladimir Kruchinin, Sep 09 2010 */

my(x='x+O('x^40)); Vec((1-x^2)/(1-2*x-x^2+x^3)) \\ G. C. Greubel, May 09 2019

{a(n) = [0, 1, 0] * [1, 1, 1; 1, 1, 0; 1, 0, 0]^(n+1) * [0, 1, 0]~}; /* Michael Somos, Dec 12 2023 */

((1-x^2)/(1-2*x-x^2+x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, May 09 2019

G.f.: (1 - x^2)/(1 - 2*x - x^2 + x^3).
a(n) = 2*a(n-1) + a(n-2) - a(n-3), with a(0)=1, a(1)=2, a(2)=4.
a(n) = Sum_{alpha = RootOf(1-2*x-x^2+x^3)} (1/7)*(2 + alpha)*alpha^(-1-n).
a(n) = central term in the (n+1)-th power of the 3 X 3 matrix (shown in the example of A066170): [1 1 1 / 1 1 0 / 1 0 0]. E.g. a(6) = 101 since the central term in M^7 = 101. - Gary W. Adamson, Feb 01 2004
a(n) = A006054(n+2) - A006054(n). - Vladimir Kruchinin, Sep 09 2010
a(n) = A077998(n+2) - 2*A006054(n+2), which implies 7*a(n-2) = (2 + c(4) - 2*c(2))*(1 + c(1))^n + (2 + c(1) - 2*c(4))*(1 + c(2))^n + (2 + c(2) - 2*c(1))*(1 + c(4))^n, where c(j)=2*Cos(2Pi*j/7), a(-2)=a(-1)=1 since A077998 and A006054 are equal to the respective quasi-Fibonacci numbers. [Witula, Slota and Warzynski] - Roman Witula, Aug 07 2012
a(n+1) = A033303(n+1) - A033303(n). - Roman Witula, Sep 14 2012
a(n) = A006054(n+2)-A006054(n). - R. J. Mathar, Nov 23 2020
a(n) = A028495(-1-n) for all n in Z. - Michael Somos, Dec 12 2023

A030116 Number of distributive lattices; also number of paths with n turns when light is reflected from 12 glass plates.

Original entry on oeis.org

1, 12, 78, 650, 5083, 40690, 323401, 2576795, 20514715, 163369570, 1300879372, 10358963615, 82488063476, 656851828075, 5230500095281, 41650400765615, 331661528811227, 2641015991983270, 21030372117368865, 167464549591889570, 1333517788817519126

Offset: 0

Jacques Haubrich (jhaubrich(AT)freeler.nl)

nonn

Let M(12) be the 12 X 12 matrix (0,0,0,1)/(0,0,1,1)/(0,1,1,1)/(1,1,1,1) and let v(12) be the 12-vector (1,1,..,1,1,1); then v(12)*M(12)^n = (x(1),x(2),...x(11),a(n)) - Benoit Cloitre, Sep 29 2002

J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124.
J. Haubrich, Multinacci Rijen [Multinacci sequences], Euclides (Netherlands), Vol. 74, Issue 4, 1998, pp. 131-133.

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]
G. Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30.
Index entries for linear recurrences with constant coefficients, signature (6, 21, -35, -70, 56, 84, -36, -45, 10, 11, -1, -1).

See also A006356-A006359, A025030, A030112-A030115.

k=12; M(k)=matrix(k,k,i,j,if(1-sign(i+j-k),0,1)); v(k)=vector(k,i,1); a(n)=vecmax(v(k)*M(k)^n)

G.f.: 1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1/(-x- 1/(-x-1/(-x-1)))))))))))). [Paul Barry, Mar 24 2010]

More terms from Benoit Cloitre, Sep 29 2002

A006360 Antichains (or order ideals) in the poset 223n or size of the distributive lattice J(223n).

Original entry on oeis.org

1, 50, 887, 8790, 59542, 307960, 1301610, 4701698, 14975675, 43025762, 113414717, 277904900, 639562508, 1393844960, 2896063220, 5768600412, 11066514565, 20526933442, 36936277875, 64660182026, 110394412610

Offset: 0

N. J. A. Sloane

J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124.
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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]
G. Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 9.
Index entries for sequences related to posets.

Cf. A000217, A000330, A050446, A050447, A006356, A006357, A006358, A006359, A000372, A056932, A006361, A006362, A056933, A056934, A056935, A056936, A056937.

Empirical G.f.: (x+1)*(x^6+36*x^5+279*x^4+594*x^3+279*x^2+36*x+1)/(1-x)^13. - Colin Barker, May 29 2012

More terms from Mitch Harris, Jul 16 2000

cp's OEIS Frontend

A056932 Antichains (or order ideals) in the poset 2*2*2*n or size of the distributive lattice J(2*2*2*n).

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A212232 T(n,k)=Number of 0..2 arrays of length n+2*k-1 with sum no more than 2*k in any length 2k subsequence (=50% duty cycle).

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A213464 T(n,k)=Number of 0..3 arrays of length n+2*k-1 with sum less than 3*k in any length 2k subsequence (=less than 50% duty cycle).

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A006358 Number of distributive lattices; also number of paths with n turns when light is reflected from 5 glass plates.

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A038196 3-wave sequence starting with 1, 1, 1.

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A025030 Number of distributive lattices; also number of paths with n turns when light is reflected from 7 glass plates.

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A030112 Number of distributive lattices; also number of paths with n turns when light is reflected from 8 glass plates.

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A056932 Antichains (or order ideals) in the poset 222n or size of the distributive lattice J(222n).

A212232 T(n,k)=Number of 0..2 arrays of length n+2k-1 with sum no more than 2k in any length 2k subsequence (=50% duty cycle).

A213464 T(n,k)=Number of 0..3 arrays of length n+2k-1 with sum less than 3k in any length 2k subsequence (=less than 50% duty cycle).

A052534 Expansion of (1-x)(1+x)/(1-2x-x^2+x^3).

A006360 Antichains (or order ideals) in the poset 223n or size of the distributive lattice J(223n).