Daniel E. Loeb - cp's OEIS frontend

A178171 Number of collections of nonempty subsets of an n-element set where each element appears in at most 3 subsets.

1, 2, 8, 109, 3623, 200522, 16514461, 1912959395, 298569495981, 60701549078701, 15647889334180500, 5003666238486522124, 1948975409748003520112, 910680909359710587298621, 503845222094502583681150340, 326363222435413478204610417626, 245078255691857705139839897934085

Offset: 0

Daniel E. Loeb, Dec 17 2010

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..100

Row n=3 of A330964.

At most 1 subset gives Bell numbers A000110, at most 2 subsets gives A178165.

a(7)-a(8) from Bert Dobbelaere, Sep 10 2019

Terms a(9) and beyond from Andrew Howroyd, Jan 04 2020

A178165 Number of unordered collections of distinct nonempty subsets of an n-element set where each element appears in at most 2 subsets.

Original entry on oeis.org

1, 2, 8, 59, 652, 9736, 186478, 4421018, 126317785, 4260664251, 166884941780, 7489637988545, 380861594219460, 21739310882945458, 1381634777325000263, 97089956842985393297, 7497783115765911443879, 632884743974716421132084

Offset: 0

Daniel E. Loeb, Dec 16 2010

nonn

If each element must appear in exactly 1 subset, then we get the Bell numbers A000110.

If each element must appear in exactly 2 subsets, then we get A002718.

Alois P. Heinz, Table of n, a(n) for n = 0..300

Row n=2 of A330964.

Cf. A094574, A000110, A002718, A178171, A178173.

terms = m = 30;
a094577[n_] := Sum[Binomial[n, k]*BellB[2n-k], {k, 0, n}];
egf = Exp[(1 - Exp[x])/2]*Sum[a094577[n]*(x/2)^n/n!, {n, 0, m}] + O[x]^m;
A094574 = CoefficientList[egf + O[x]^m, x]*Range[0, m-1]!;
a[n_] := Sum[Binomial[n, k]*A094574[[k+1]], {k, 0, n}];
Table[a[n], {n, 0, m-1}] (* Jean-François Alcover, May 24 2019 *)

from numpy import array
def toBinary(n, k):
    ans=[]
    for i in range(k):
        ans.insert(0, n%2)
        n=n>>1
    return array(ans)
def powerSet(k): return [toBinary(n,k) for n in range(1,2**k)]
def courcelle(maxUses, remainingSets, exact=False):
    if exact and not all(maxUses<=sum(remainingSets)): ans=0
    elif len(remainingSets)==0: ans=1
    else:
        set0=remainingSets[0]
        if all(set0<=maxUses): ans=courcelle(maxUses-set0,remainingSets[1:],exact=exact)
        else: ans=0
        ans+=courcelle(maxUses,remainingSets[1:],exact=exact)
    return ans
for i in range(10):
    print(i, courcelle(array([2]*i),powerSet(i),exact=False))

Binomial transform of A094574: a(n) = Sum_{k=0..n} C(n,k)*A094574(k).

Edited and corrected by Max Alekseyev, Dec 19 2010

A178173 Number of collections of nonempty subsets of an n-element set where each element appears in at most 4 subsets.

Original entry on oeis.org

1, 2, 8, 128, 11087, 2232875, 775098224, 428188962261, 355916994389700, 425272149099677521, 703909738878615927739, 1565842283246869237505246, 4565002967677134523844716754, 17076464900445281560851997140670, 80494979734877344662882495100752511

Offset: 0

Daniel E. Loeb, Dec 17 2010

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..50

Row n=4 of A330964.

Replacing limit of 2 with a limit of 1 gives the Bell numbers A000110, limit of 2 gives A178165, limit of 3 gives A178171.

\\ See A330964 for efficient code to compute this sequence. - Andrew Howroyd, Jan 04 2020

from numpy import array
def toBinary(n,k):
    ans=[]
    for i in range(k):
        ans.insert(0,n%2)
        n=n>>1
    return array(ans)
def powerSet(k): return [toBinary(n,k) for n in range(1,2**k)]
def courcelle(maxUses,remainingSets,exact=False):
    if exact and not all(maxUses<=sum(remainingSets)): ans=0
    elif len(remainingSets)==0: ans=1
    else:
        set0=remainingSets[0]
        if all(set0<=maxUses): ans=courcelle(maxUses-set0,remainingSets[1:],exact=exact)
        else: ans=0
        ans+=courcelle(maxUses,remainingSets[1:],exact=exact)
    return ans
for i in range(10):
    print(i, courcelle(array([4]*i),powerSet(i),exact=False))

a(6)-a(8) from Bert Dobbelaere, Sep 10 2019

Terms a(9) and beyond from Andrew Howroyd, Jan 04 2020

A176341 a(n) = the location of the first appearance of the decimal expansion of n in the decimal expansion of Pi.

Original entry on oeis.org

32, 1, 6, 0, 2, 4, 7, 13, 11, 5, 49, 94, 148, 110, 1, 3, 40, 95, 424, 37, 53, 93, 135, 16, 292, 89, 6, 28, 33, 186, 64, 0, 15, 24, 86, 9, 285, 46, 17, 43, 70, 2, 92, 23, 59, 60, 19, 119, 87, 57, 31, 48, 172, 8, 191, 130, 210, 404, 10, 4, 127, 219, 20, 312, 22, 7, 117, 98, 605, 41

Offset: 0

Daniel E. Loeb, Apr 15 2010

It is unknown whether Pi is a normal number. If it is (at least in base 10) then this sequence is well defined.

The numbers a(n) refer to the position of the initial digit of n in the decimal expansion of Pi, where "3" is at position a(3)=0, "1" is at position a(1)=1, etc. This is also the numbering scheme used on the "Pi search page" cited among the LINKS. See A232013 for a sequence based on iterations of this one. See A032445 for a variant of the present sequence, where numbering starts at one. - M. F. Hasler, Nov 16 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Dave Andersen, Search within first 200,000,000 digits of pi
Michael D. Huberty, Ko Hayashi & Chia Vang, First 100,000 digits of pi
Simon Plouffe, First 50,000,000 digits of pi

Cf. A000796, A011545, A014777, A032445, A232013.

p=ToString[FromDigits[RealDigits[N[Pi, 10^4]][[1]]]]; Do[Print[StringPosition[p, ToString[n]][[1]][[1]] - 1], {n, 0, 100}] (* Vincenzo Librandi, Apr 17 2017 *)
With[{pid=RealDigits[Pi,10,800][[1]]},Flatten[Table[ SequencePosition[ pid,IntegerDigits[n],1],{n,0,70}],1]][[All,1]]-1 (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 27 2019 *)

A176341(n)=my(L=#Str(n));n=Mod(n,10^L);for(k=L-1,9e9,Pi\.1^k-n||return(k+1-L)) \\ Make sure to use sufficient realprecision, e.g. via \p999. - M. F. Hasler, Nov 16 2013

pi = "314159265358979323846264338327950288419716939937510582097494459230..."
[ pi.find(str(i)) for i in range(10000) ]

a(n) = A032445(n)-1. - M. F. Hasler, Nov 16 2013

a(n) = 0 if n is in A011545, otherwise a(n) = A014777(n). - Pontus von Brömssen, Aug 31 2024

A018223 Number of 5-voter voting schemes with n linearly ranked choices.

Original entry on oeis.org

1, 81, 2646, 43556, 476120, 3785410

Offset: 1

Daniel E. Loeb

nonn

Daniel E. Loeb, On Games, Voting Schemes and Distributive Lattices. LaBRI Report 625-93, University of Bordeaux I, 1993. [broken link]

A008842 Number of inequivalent self-dual Boolean functions of n variables with transitive symmetry group.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 1, 2, 0, 24, 28, 57259, 0

Offset: 0

Daniel E. Loeb

Index entries for sequences related to Boolean functions

A008841 Number of self-dual Boolean functions of n variables with transitive symmetry group.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 12, 31, 0, 570361, 1441440, 207648650161, 0

Offset: 0

Daniel E. Loeb

Index entries for sequences related to Boolean functions

A008840 Number of monotone self-dual Boolean functions of n variables that are inequivalent under the symmetric group.

Original entry on oeis.org

0, 1, 1, 2, 3, 7, 30, 716

Offset: 0

Daniel E. Loeb

D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.1, p. 79.

Daniel Elliott Loeb, Home page
Bartłomiej Pawelski, Counting and generating monotone Boolean functions, Doctoral Diss., Univ. Gdańsk, (Poland, 2024). See pp. 12, 15, 58, 67, 73. (Abstract in Polish.)
Index entries for sequences related to Boolean functions

Cf. A003180, A057132, A108803.

Cf. A107765 (non-isomorphic self dual monotones = differences of A008840?). - Olivier Gérard, Oct 11 2012

A007007 Valence of graph of maximal intersecting families of sets.

Original entry on oeis.org

0, 1, 3, 4, 10, 15, 35, 56, 126, 210, 462, 792, 1716, 3003, 6435, 11440, 24310, 43758, 92378, 167960, 352716, 646646, 1352078, 2496144, 5200300, 9657700, 20058300, 37442160, 77558760, 145422675, 300540195, 565722720, 1166803110, 2203961430

Offset: 1

Daniel E. Loeb

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

Jan C. Bioch and Toshihide Ibaraki, Generating and approximating nondominated coteries, IEEE Transactions on parallel and distributed systems 6 (1995), 905-914.
D. E. Loeb and A. Meyerowitz, The maximal intersecting family of sets graph, in H. Barcelo and G. Kalai, editors, Proceedings of the Conference on Jerusalem Combinatorics 1993. AMS series Contemporary Mathematics, 1994.
A. Meyerowitz, Maximal intersecting families, European J. Combin. 16 (1995), no. 5, 491-501.

Cf. A001206, A007006, A007008.

Equals A037952 except for n=1 [Loeb & Meyerowitz, theorem 16]. - Andrey Zabolotskiy, Sep 19 2017

More terms from Andrey Zabolotskiy, Sep 19 2017

A007010 Number of 4-voter voting schemes with n linearly ranked choices.

Original entry on oeis.org

1, 12, 81, 372, 1332, 3984, 10420, 24540, 53145, 107436, 205065, 372792, 649936, 1092672, 1779408, 2817288, 4350105, 6567660, 9716905, 14114892, 20163924, 28368912, 39357396, 53902212, 72947329, 97636812, 129347505, 169725360, 220726080, 284659968, 364241728

Offset: 1

Daniel E. Loeb

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

Colin Barker, Table of n, a(n) for n = 1..1000
Daniel E. Loeb, On Games, Voting Schemes and Distributive Lattices. LaBRI Report 625-93, University of Bordeaux I, 1993. [broken link]
Index entries for linear recurrences with constant coefficients, signature (6,-12,2,27,-36,0,36,-27,-2,12,-6,1).

LinearRecurrence[{6,-12,2,27,-36,0,36,-27,-2,12,-6,1},{1,12,81,372,1332,3984,10420,24540,53145,107436,205065,372792},40] (* Harvey P. Dale, Feb 12 2023 *)

Vec(x*(1+6*x+21*x^2+28*x^3+21*x^4+6*x^5+x^6)/((1+x)^3*(1-x)^9) + O(x^100)) \\ Colin Barker, Jan 07 2016

G.f.: x*(1+6*x+21*x^2+28*x^3+21*x^4+6*x^5+x^6)/((1+x)^3*(1-x)^9). - Ralf Stephan, Apr 23 2004
From Colin Barker, Jan 07 2016: (Start)
a(n) = (n^8+16*n^7+106*n^6+376*n^5+784*n^4+1024*n^3+864*n^2+384*n)/3840 for n even.
a(n) = (n^8+16*n^7+106*n^6+376*n^5+784*n^4+1024*n^3+894*n^2+504*n+135)/3840 for n odd.
(End)

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User: Daniel E. Loeb

A178171 Number of collections of nonempty subsets of an n-element set where each element appears in at most 3 subsets.

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A178165 Number of unordered collections of distinct nonempty subsets of an n-element set where each element appears in at most 2 subsets.

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A178173 Number of collections of nonempty subsets of an n-element set where each element appears in at most 4 subsets.

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A176341 a(n) = the location of the first appearance of the decimal expansion of n in the decimal expansion of Pi.

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PARI

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A018223 Number of 5-voter voting schemes with n linearly ranked choices.

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A008842 Number of inequivalent self-dual Boolean functions of n variables with transitive symmetry group.

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A008841 Number of self-dual Boolean functions of n variables with transitive symmetry group.

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A008840 Number of monotone self-dual Boolean functions of n variables that are inequivalent under the symmetric group.

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A007007 Valence of graph of maximal intersecting families of sets.

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A007010 Number of 4-voter voting schemes with n linearly ranked choices.

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