A006356 -id:A006356 - cp's OEIS frontend

A006361 Antichains (or order ideals) in the poset 224n or size of the distributive lattice J(224n).

1, 105, 3490, 59542, 650644, 5157098, 32046856, 164489084, 723509159, 2801747767, 9748942554, 30967306114, 90930233726, 249319296218, 643622467414, 1575086681342, 3675063064675, 8215220917795

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.
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. A000372, A056932, A006360, A006362, A056933, A056934, A056935, A056936, A056937.

Empirical G.f.: (x^10 +88*x^9 +1841*x^8 +13812*x^7 +44050*x^6 +64374*x^5 +44050*x^4 +13812*x^3 +1841*x^2 +88*x +1)/(1-x)^17. - Colin Barker, May 29 2012

More terms from Mitch Harris, Jul 16 2000

A006362 Antichains (or order ideals) in the poset 225n or size of the distributive lattice J(225n).

Original entry on oeis.org

1, 196, 11196, 307960, 5157098, 60112692, 530962446, 3764727340, 22326282261, 114158490576, 515063238810, 2087929609236, 7714649716552, 26285397502428, 83379798088110, 248202756212336, 697998155989501, 1864955619299196

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.
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.
Index entries for sequences related to posets.

Cf. A000372, A056932, A006360, A006361, A056933, A056934, A056935, A056936, A056937.

More terms from Mitch Harris, Jul 16 2000

A056933 Antichains (or order ideals) in the poset 226n or size of the distributive lattice J(226n).

Original entry on oeis.org

1, 336, 30900, 1301610, 32046856, 530962446, 6479344016, 61951251333, 485198553532, 3217462615688, 18528857431906, 94529315562186, 434088353496446, 1817613939845670, 7014049051387480, 25167786776727516

Offset: 0

Mitch Harris

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

J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]
Index entries for sequences related to posets

Cf. A000372, A006360, A006361, A006362, A056932, A056934, A056935, A056936, A056937.

A056934 Antichains (or order ideals) in the poset 227n or size of the distributive lattice J(227n).

Original entry on oeis.org

1, 540, 75966, 4701698, 164489084, 3764727340, 61951251333, 782318812002, 7946895019096, 67270102239520, 487605585591870, 3092040981805272, 17451258588313354, 88902214572208640, 413569247116248032

Offset: 0

Mitch Harris

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

J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]
Index entries for sequences related to posets

Cf. A000372, A006360, A006361, A006362, A056932, A056933, A056935, A056936, A056937.

A056935 Antichains (or order ideals) in the poset 233n or size of the distributive lattice J(233n).

Original entry on oeis.org

1, 175, 8790, 211250, 3092808, 31635580, 246441430, 1549486490, 8192995479, 37548347569, 152602439244, 559835910940, 1880152558980, 5846268790220, 16988036626948, 46486024648180, 120562654732065, 297976456047575

Offset: 0

Mitch Harris

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

J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]
Index entries for sequences related to posets

Cf. A000372, A006360, A006361, A006362, A056932, A056933, A056934, A056936, A056937.

A056936 Antichains (or order ideals) in the poset 234n or size of the distributive lattice J(234n).

Original entry on oeis.org

1, 490, 59542, 3092808, 89613429, 1691136270, 22954776044, 239831111938, 2024703039198, 14318216628378, 87184226214168, 467034400160664, 2239064967256060, 9741467994941264, 38902816410160608

Offset: 0

Mitch Harris

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

J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]
Index entries for sequences related to posets

Cf. A000372, A006360, A006361, A006362, A056932, A056933, A056934, A056935, A056937.

A056937 Number of antichains (or order ideals) in the poset 333n or size of the distributive lattice J(333n).

Original entry on oeis.org

1, 980, 211250, 17792748, 781429368, 21238316210, 398925639186, 5585711269074, 61555624183223, 555895303974238, 4242859829536322, 28038281717424550, 163544036697306396, 855242362045150398

Offset: 0

Mitch Harris

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

J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]
Index entries for sequences related to posets

Cf. A000372, A006360, A006361, A006362, A056932, A056933, A056934, A056935, A056936.

A296834 T(n,k)=Number of nXk 0..1 arrays with each 1 adjacent to 0, 3 or 5 king-move neighboring 1s.

Original entry on oeis.org

2, 3, 3, 5, 6, 5, 8, 14, 14, 8, 13, 31, 43, 31, 13, 21, 70, 132, 132, 70, 21, 34, 157, 402, 573, 402, 157, 34, 55, 353, 1230, 2441, 2441, 1230, 353, 55, 89, 793, 3755, 10485, 14379, 10485, 3755, 793, 89, 144, 1782, 11475, 44951, 85500, 85500, 44951, 11475, 1782, 144

Offset: 1

R. H. Hardin, Dec 21 2017

Table starts
..2....3.....5......8.......13........21.........34...........55............89
..3....6....14.....31.......70.......157........353..........793..........1782
..5...14....43....132......402......1230.......3755........11475.........35054
..8...31...132....573.....2441.....10485......44951.......192730........826498
.13...70...402...2441....14379.....85500.....508111......3017667......17931240
.21..157..1230..10485....85500....706534....5834429.....48122349.....397252886
.34..353..3755..44951...508111...5834429...67007971....768117235....8815633972
.55..793.11475.192730..3017667..48122349..768117235..12228697715..194991656916
.89.1782.35054.826498.17931240.397252886.8815633972.194991656916.4321362259224

			Some solutions for n=5 k=4
..0..0..1..1. .0..0..0..0. .1..0..0..0. .0..1..0..1. .0..0..0..1
..1..0..1..1. .1..1..1..1. .0..0..1..0. .0..0..0..0. .0..1..0..0
..0..0..0..0. .1..1..1..1. .1..0..0..0. .0..0..0..0. .0..0..0..0
..0..0..0..0. .0..0..0..0. .0..0..1..1. .1..0..0..1. .0..1..0..0
..0..0..1..0. .0..0..0..1. .0..0..1..1. .0..0..0..0. .0..0..0..1

R. H. Hardin, Table of n, a(n) for n = 1..199

Column 1 is A000045(n+2).

Column 2 is A006356.

Empirical for column k:
k=1: a(n) = a(n-1) +a(n-2)
k=2: a(n) = 2*a(n-1) +a(n-2) -a(n-3)
k=3: [order 15]
k=4: [order 50]

A106803 Expansion of x(1-x)/(1-2x-x^2+x^3).

Original entry on oeis.org

0, 1, 1, 3, 6, 14, 31, 70, 157, 353, 793, 1782, 4004, 8997, 20216, 45425, 102069, 229347, 515338, 1157954, 2601899, 5846414, 13136773, 29518061, 66326481, 149034250, 334876920, 752461609, 1690765888, 3799116465, 8536537209

Offset: 0

Roger L. Bagula, May 17 2005

Essentially a duplicate of A077998: a(n) = A077998(n-1). - Joerg Arndt, Aug 14 2015
a(n) appears in the formula for the nonnegative powers of sigma, the ratio of the smaller diagonal in the heptagon to the side length s=2*sin(Pi/7), when expressed in the basis <1,rho,sigma>, with rho = 2*cos(Pi/7), the ratio of the smaller heptagon diagonal to the side length, as follows. sigma^n = a(n-1)*1 + B(n)*rho + a(n)*sigma, n>=0, with B(n)=A006054(n). Put a(-1):= 1. See the Steinbach reference, and a comment under A052547.
a(n-1) is the top left entry of the n-th power of the 3X3 matrix [0, 1, 0; 1, 1, 1; 0, 1, 1] or of the 3X3 matrix [0, 0, 1; 0, 1, 1; 1, 1, 1]. - R. J. Mathar, Feb 03 2014

Michael De Vlieger, Table of n, a(n) for n = 0..2845
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), p. 22-31.
Kai Wang, Fibonacci Numbers And Trigonometric Functions Outline, (2019).
Index entries for linear recurrences with constant coefficients, signature (2,1,-1).

Cf. A006356, A077998, A187068, A187070, A199853.

m = {{0, 0, 1}, {1, 2, 0}, {1, 1, 0}}; v[0] = {0, 1, 1}; v[n_] := m.v[n - 1]; Table[v[n][[1]], {n, 0, 30}] (* Edited and corrected by L. Edson Jeffery, Oct 18 2017 *)
RecurrenceTable[{a[1]== 0, a[2]== 1, a[3]== 1, a[n]== 2*a[n-1]  + a[n-2] - a[n-3]}, a, {n,30}] (* G. C. Greubel, Aug 14 2015 *)

concat(0,Vec((1-x)/(x^3-2*x-x^2+1)+O(x^99))) \\ Charles R Greathouse IV, Sep 25 2012

a(n) = A077998(n-1). - R. J. Mathar, Aug 07 2008
a(n) = A187070(2*n), a(n) = A187068(2*n+2). - L. Edson Jeffery, Mar 10 2011
a(n+1) = - A199853(n+1). - G. C. Greubel, Aug 14 2015
a(n) = 2*a(n-1) + a(n-2) - a(n-3), a(0)=0, a(1)=a(2)=1. - G. C. Greubel, Aug 14 2015
a(n) = A006356(n-2) for n > 1. - Georg Fischer, Oct 21 2018

Edited by N. J. A. Sloane, Aug 08 2008

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

cp's OEIS Frontend

A006361 Antichains (or order ideals) in the poset 2*2*4*n or size of the distributive lattice J(2*2*4*n).

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A006362 Antichains (or order ideals) in the poset 2*2*5*n or size of the distributive lattice J(2*2*5*n).

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A056933 Antichains (or order ideals) in the poset 2*2*6*n or size of the distributive lattice J(2*2*6*n).

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A056934 Antichains (or order ideals) in the poset 2*2*7*n or size of the distributive lattice J(2*2*7*n).

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A056935 Antichains (or order ideals) in the poset 2*3*3*n or size of the distributive lattice J(2*3*3*n).

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A056936 Antichains (or order ideals) in the poset 2*3*4*n or size of the distributive lattice J(2*3*4*n).

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A056937 Number of antichains (or order ideals) in the poset 3*3*3*n or size of the distributive lattice J(3*3*3*n).

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A296834 T(n,k)=Number of nXk 0..1 arrays with each 1 adjacent to 0, 3 or 5 king-move neighboring 1s.

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Formula

A106803 Expansion of x*(1-x)/(1-2*x-x^2+x^3).

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Mathematica

PARI

Formula

Extensions

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

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Examples

A006361 Antichains (or order ideals) in the poset 224n or size of the distributive lattice J(224n).

A006362 Antichains (or order ideals) in the poset 225n or size of the distributive lattice J(225n).

A056933 Antichains (or order ideals) in the poset 226n or size of the distributive lattice J(226n).

A056934 Antichains (or order ideals) in the poset 227n or size of the distributive lattice J(227n).

A056935 Antichains (or order ideals) in the poset 233n or size of the distributive lattice J(233n).

A056936 Antichains (or order ideals) in the poset 234n or size of the distributive lattice J(234n).

A056937 Number of antichains (or order ideals) in the poset 333n or size of the distributive lattice J(333n).

A106803 Expansion of x(1-x)/(1-2x-x^2+x^3).