Jan Kristian Haugland - cp's OEIS frontend

A346848 Number of conjugacy classes of the symplectic group Sp(2n, 2) over the field with 2 elements.

1, 3, 11, 30, 81, 198, 477, 1089, 2451, 5358, 11567, 24537, 51577, 107205, 221378, 453900, 926395, 1882152, 3812232, 7699191, 15518112, 31220991, 62733296, 125911851, 252516626, 506082933, 1013780968, 2029989807, 4063678159, 8132877129, 16274093175

Offset: 0

Jan Kristian Haugland, Aug 06 2021

nonn

Sp(2n, 2) is isomorphic to the orthogonal group O(2n+1, 2) over the field with 2 elements, and is a simple and complete group for n>=3.

			a(2)=11, and Sp(4, 2) is isomorphic to the symmetric group S_6 which has 11 conjugacy classes.

Jan Kristian Haugland, Table of n, a(n) for n = 0..1000
G. E. Wall, On the conjugacy classes in the unitary, symplectic and orthogonal groups, J. Aust. Math. Soc., 3 (1963), 1-62.

Discrete convolution of A070933 and A098613. A003923 gives the order of the group.

A308489 Number of 5-regular polyhedra with 2n nodes.

Original entry on oeis.org

1, 0, 1, 1, 6, 14, 96, 518, 3917, 29821, 240430, 1957382, 16166596

Offset: 6

Jan Kristian Haugland, Jun 12 2019

Number of simple 5-regular 3-connected planar graphs with 2n nodes.

Mahdieh Hasheminezhad, Brendan D. McKay, and Tristan Reeves, Recursive generation of 5-regular planar graphs, WALCOM: Algorithms and Computation: Third International Workshop, WALCOM 2009, Kolkata, India, February 18-20, 2009, Proceedings, Springer, 2009, pp. 129-140.

Cf. A000944, A301425.

A308588 Number of 4-regular 3-connected planar graphs with n vertices having an Eulerian tour for which no two consecutive edges are incident with the same face.

Original entry on oeis.org

1, 1, 1, 1, 3, 5, 17, 40, 145, 355, 1264, 3931, 12999, 44727

Offset: 8

Jan Kristian Haugland, Jun 09 2019

Polyhedra formed by knot diagrams.

Stuart E Anderson, Illustrations of a(8) to a(14) 4-regular 3-connected planar graphs with n vertices having an Eulerian tour for which no two consecutive edges are incident with the same face.

Cf. A007022.

A144188 a(n)/n! is the probability of guessing "up/down" correctly through a deck of n cards marked 1, 2, ..., n, if one always makes the most probable guess.

Original entry on oeis.org

1, 1, 2, 5, 16, 62, 286, 1519, 9184, 62000, 463964, 3800684, 33911424, 326678010, 3385261194, 37492199549, 442541571936, 5539379635136, 73368335117584, 1024178393797764, 15041551052243448, 231665680071392900, 3736363255881557460, 62935656581952683960

Offset: 0

Jan Kristian Haugland, Sep 13 2008

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..474 (first 31 terms from Jan Kristian Haugland)

Let f(0, 0) = 1 and f(n, k) = max{f(n - 1, 0) + ... + f(n - 1, k - 1), f(n - 1, k) + ... + f(n - 1, n - 1)} for 0 <= k <= n. Then a(n) = f(n, 0).

A130695 Number of ways to write n as (a+1)(b+1)(c+1) - abc with a, b, c nonnegative integers.

Original entry on oeis.org

1, 3, 3, 6, 3, 9, 4, 9, 6, 12, 3, 15, 6, 12, 9, 15, 3, 21, 7, 15, 9, 18, 6, 24, 9, 15, 9, 24, 6, 30, 6, 15, 15, 24, 9, 30, 7, 21, 12, 30, 3, 33, 15, 21, 15, 24, 6, 39, 12, 27, 12, 27, 9, 42, 12, 21, 15, 36, 6, 45, 13, 18, 21, 36, 12, 39, 6, 33, 15, 45, 9, 42, 12, 24, 24, 30, 9, 57, 18, 30

Offset: 1

Jan Kristian Haugland, Jul 10 2007

nonn

			a(7) = 4 because 7 = 7*1*1-6*0*0 = 1*7*1-0*6*0 = 1*1*7-0*0*6 = 2*2*2-1*1*1.
G.f. = x + 3*x^2 + 3*x^3 + 6*x^4 + 3*x^5 + 9*x^6 + 4*x^7 + 9*x^8 + 6*x^9 + ...

_Jan Kristian Haugland_, Jul 10 2007, Table of n, a(n) for n = 1..200

Cf. A238872.

f[{a_,b_,c_}]:=(a+1)(b+1)(c+1)-a*b*c; nn=80;Take[Transpose[Sort[Tally[f/@ Tuples[Range[0,nn],3]],#1[[1]]<#2[[1]]&]] [[2]],nn] (* Harvey P. Dale, Mar 05 2012 *)
a[ n_] := Length @ FindInstance[ {x >= 0, y >= 0, z >= 0, x y + y z + z x + x + y + z + 1 == n}, {x, y, z}, Integers, 10^9]; (* Michael Somos, Jul 04 2015 *)
a[ n_] := (2 + (-1)^n) Length @ FindInstance[ {1 <= y <= n, 1 <= x <= y, 1 <= z <= y, y^2 - (x^2 - x + z^2 - z) / 2 == n}, {x, y, z}, Integers, 10^9]; (* Michael Somos, Jul 04 2015 *)

a(2*n) = A238872(2*n) / 3 if n>0. a(2*n + 1) = A238872(2*n + 1). - Michael Somos, Jul 04 2015

A094777 Number of legal positions in Go played on an n X n grid (each group must have at least one liberty).

Original entry on oeis.org

1, 57, 12675, 24318165, 414295148741, 62567386502084877, 83677847847984287628595, 990966953618170260281935463385, 103919148791293834318983090438798793469, 96498428501909654589630887978835098088148177857, 793474866816582266820936671790189132321673383112185151899, 57774258489513238998237970307483999327287210756991189655942651331169, 37249792307686396442294904767024517674249157948208717533254799550970595875237705, 212667732900366224249789357650440598098805861083269127196623872213228196352455447575029701325

Offset: 1

Jan Kristian Haugland, Jun 09 2004

nonn

John Tromp wrote a small C program to compute the number for boards up to size 4 X 5, given in the rec.games.go posting below. Gunnar Farnebaeck (gunnar(AT)lysator.liu.se) wrote a pike script to compute the number by dynamic programming, which handles sizes up to 12 X 12 (available upon request).

			The illegal 2 X 2 positions are the 2^4 with no empty points and the 4*2 having a stone adjacent to 2 opponent stones that share a liberty. That leaves 3^4-16-8 = 57 legal positions.

John Tromp, Table of n, a(n) for n = 1..19
British Go Association, Go
Sandy Harris, Number of Possible Outcomes of a Game
John Tromp, Complexity of Chess and Go
John Tromp, Number of legal Go positions
John Tromp and Gunnar Farnebäck, Combinatorics of Go (2016)
Zach Weinersmith, A New Branch of Mathematics, SMBC (Saturday Morning Breakfast Cereal) column, Mar 24 2025 [Mentions this sequence] (Link provided by _Jeffrey Shallit_, Mar 24 2025)

3^(n*n) is a trivial upper bound.

Tromp & Farnebäck prove that a(n) = (1 + o(1)) * L^(n^2), and conjecture that a(n) ~ A * B^(2n) * L^(n^2) * (1 + O(n*p^n)) for some constants A, B, L, and p < 1. - Charles R Greathouse IV, Feb 08 2016

More terms from John Tromp, Jan 27 2005
a(10)-a(13) from John Tromp, Jun 23 2005
a(14)-a(15) from John Tromp, Sep 01 2005
a(16) from John Tromp, Oct 06 2005
Michal Koucky should be credited for carrying most of the computational load for computing the n=14, 15 and 16 results with his file-based implementation.
a(17)-a(18) from John Tromp, Mar 08 2015
a(19) from John Tromp, Jan 21 2016

A066369 Number of subsets of {1, ..., n} with no four terms in arithmetic progression.

Original entry on oeis.org

1, 2, 4, 8, 15, 29, 56, 103, 192, 364, 668, 1222, 2233, 3987, 7138, 12903, 22601, 40200, 71583, 125184, 218693, 386543, 670989, 1164385, 2021678, 3462265, 5930954, 10189081, 17266616, 29654738, 50912618, 86017601, 145327544, 247555043, 415598432, 698015188

Offset: 0

Jan Kristian Haugland, Dec 22 2001

nonn

			a(5) = 29 because there are 32 subsets and three of them contain four terms in arithmetic progression: {1, 2, 3, 4}, {2, 3, 4, 5} and {1, 2, 3, 4, 5}.

Cf. A051013, A018789.

from sympy import subsets
def noap4(n):
    avoid=list()
    for skip in range(1,(n+2)//3):
        for start in range (1,n+1-3*skip):
            avoid.append(set({start,start+skip,start+2*skip,start+3*skip}))
    s=list()
    for i in range(4):
        for smallset in subsets(range(1,n+1),i):
            s.append(smallset)
    for i in range(4,n+1):
        for temptuple in subsets(range(1,n+1),i):
            tempset=set(temptuple)
            status=True
            for avoidset in avoid:
                if avoidset <= tempset:
                    status=False
                    break
            if status:
                s.append(tempset)
    return s
# Counts all such sets
def a(n):
    return len(noap4(n)) # David Nacin, Mar 05 2012

a(n) = 2^n - A018789(n).

a(31)-a(35) (using data in A018789) from Alois P. Heinz, Sep 08 2019

A066331 Number of fixed hexagonal polyominoes with n cells and tree-like structure.

Original entry on oeis.org

1, 3, 9, 29, 99, 348, 1260, 4644, 17382, 65822, 251655, 969819, 3762517, 14680890, 57567228, 226712655, 896252850, 3555116583, 14144563158

Offset: 1

Jan Kristian Haugland, Jan 01 2002

Sean A. Irvine, Java program (github)

Cf. A001207 (all polyhexes), A038142 (free tree-like).

a(16)-a(19) from Sean A. Irvine, Oct 08 2023

A065068 Number of fixed polyominoes with n cells of which no four are equally spaced on a straight line.

Original entry on oeis.org

1, 2, 6, 17, 45, 114, 264, 545, 1061, 2070, 3990, 7275, 12556, 20824, 33584, 52753, 80560, 119274, 170168, 232700, 306472, 396580, 517300, 688736, 926816, 1241720, 1640968, 2130448, 2710356, 3383596, 4171788, 5120776, 6289760, 7747928, 9582196, 11908872, 14887792, 18732340, 23712492, 30141180, 38341172, 48622328, 61275752, 76578628, 94786504, 116132688, 140870564, 169379536, 202262720, 240326780, 284337040, 334532714, 389976092

Offset: 1

Jan Kristian Haugland, Nov 07 2001

nonn

a(142) = 36 is the final nonzero term of this sequence. - Jan Kristian Haugland, Apr 28 2020

Jan Kristian Haugland, Illustration for n = 142 [Cached copy, with permission]
J. K. Haugland, Largest polyomino with no four cells equally spaced on a straight line, arXiv:2004.12801 [math.GM] (2020).
Wikipedia, Polyomino

Cf. A001168.

a(n) = 0 for n >= 143. - Jan Kristian Haugland, Apr 28 2020

a(32)-a(53) from Brendan Owen (brendan_owen(AT)yahoo.com), Mar 11 2002

A066158 Number of fixed polyominoes with n cells and tree-like structure.

Original entry on oeis.org

1, 2, 6, 18, 55, 174, 570, 1908, 6473, 22202, 76886, 268352, 942651, 3329608, 11817582, 42120340, 150682450, 540832274, 1946892842, 7027047848, 25424079339, 92185846608, 334925007128, 1219054432490, 4444545298879, 16229462702152, 59347661054364

Offset: 1

Jan Kristian Haugland, Dec 13 2001

Computed by a modified version of the program used for A065068.

Aleksandrowicz and Barequet (2011) confirm first 27 terms. - Gill Barequet, May 25 2011

G. Aleksandrowicz and G. Barequet, Parallel enumeration of lattice animals, Proc. 5th Int. Frontiers of Algorithmics Workshop, Zhejiang, China, Lecture Notes in Computer Science, 6681, Springer-Verlag, 90-99, May 2011.
N. Madras, C. E. Soteros, S. G. Whittington, J. L. Martin, M. F. Sykes et al., The free energy of a collapsing branched polymer, J. Phys. A: Math. Gen. 23 (1990) 5327-5350.

I. Jensen, Table of n, a(n) for n = 1..44 [From the arXiv paper]
Gill Barequet, Gil Ben-Shachar, Martha Carolina Osegueda, Applications of Concatenation Arguments to Polyominoes and Polycubes, EuroCG '20, 36th European Workshop on Computational Geometry, (Würzburg, Germany, 16-18 March 2020).
I. Jensen, Enumerations of lattice animals and trees, J. Stat. Phys. 103 (3-4) (2001) 865-881, Table II.
I. Jensen, Enumerations of lattice animals and trees, arXiv:cond-mat/0007239.

Cf. A001168 (fixed polyominoes), A019441 (coefficients of g.f. related to this sequence), A118356, A191094, A191095, A191096, A191097, A191098 (fixed tree-like polycubes in 3, 4, 5, 6, 7, and 8 dimensions, resp.).

Added a(18) and a(19) from Madras et al. - R. J. Mathar, Apr 08 2006

Terms from a(20) on added by N. J. A. Sloane, Nov 05 2008, from the Jensen paper.

cp's OEIS Frontend

User: Jan Kristian Haugland

A346848 Number of conjugacy classes of the symplectic group Sp(2n, 2) over the field with 2 elements.

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A308489 Number of 5-regular polyhedra with 2n nodes.

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A308588 Number of 4-regular 3-connected planar graphs with n vertices having an Eulerian tour for which no two consecutive edges are incident with the same face.

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A144188 a(n)/n! is the probability of guessing "up/down" correctly through a deck of n cards marked 1, 2, ..., n, if one always makes the most probable guess.

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A130695 Number of ways to write n as (a+1)(b+1)(c+1) - abc with a, b, c nonnegative integers.

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Mathematica

Formula

A094777 Number of legal positions in Go played on an n X n grid (each group must have at least one liberty).

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Extensions

A066369 Number of subsets of {1, ..., n} with no four terms in arithmetic progression.

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Python

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A066331 Number of fixed hexagonal polyominoes with n cells and tree-like structure.

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A065068 Number of fixed polyominoes with n cells of which no four are equally spaced on a straight line.

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Crossrefs

Formula

Extensions

A066158 Number of fixed polyominoes with n cells and tree-like structure.

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Comments