A002857 -id:A002857

Search: a002857 -id:a002857

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A002543

Complete Post functions of n variables.
(Formerly M2098 N0830)

+10
6

0, 2, 16, 980, 9332768

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

REFERENCES

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

Wheeler, Roger F.; Complete propositional connectives. Z. Math. Logik Grundlagen Math. 7 1961 185-198.

Wheeler, Roger F.; Complete connectives for the 3-valued propositional calculus. Proc. London Math. Soc. (3) 16 1966 167-191.

LINKS

Table of n, a(n) for n=1..5.

R. F. Wheeler, Complete propositional connectives, Z. Math. Logik Grundlagen Math. 7 1961 185-198. [Annotated scanned copy]

R. F. Wheeler, An asymptotic formula for the number of complete propositional connectives, Z. Math. Logik Grundlagen Math. 8 (1962), 1-4. [Annotated scanned copy]

R. F. Wheeler, Complete connectives for the 3-valued propositional calculus, Proc. London Math. Soc. (3) 16 (1966), 167-191. [Annotated scanned copy]

CROSSREFS

Cf. A002542, A002857.

KEYWORD

nonn,more

AUTHOR

N. J. A. Sloane

STATUS

approved

A055152

Proper covers of an unlabeled n-set.

+10
4

0, 1, 14, 956, 9331320, 6406603065901952, 16879085743296493569230716352778240, 717956902513121252476003434439730211883694285987816199468264943161704448

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..12

Heller, Jürgen Identifiability in probabilistic knowledge structures. J. Math. Psychol. 77, 46-57 (2017).

Eric Weisstein's World of Mathematics, Proper covers

FORMULA

a(n) = (A003180(n) - 2*A003180(n-1))/4.

Apparently a(n) = A002857(n) - A000612(n-1). - R. J. Mathar, Apr 22 2007

MAPLE

b:= proc(n, i, l) `if`(n=0, 2^(w-> add(mul(2^igcd(t, l[h]),

h=1..nops(l)), t=1..w)/w)(ilcm(l[])), `if`(i<1, 0,

add(b(n-i*j, i-1, [l[], i$j])/j!/i^j, j=0..n/i)))

end:

a:= n-> (b(n$2, [])-2*b(n-1$2, []))/4:

seq(a(n), n=1..8); # Alois P. Heinz, Aug 14 2019

MATHEMATICA

b[n_] := Sum[1/Function[p, Product[Function[c, j^c*c!][Coefficient[p, x, j]], {j, 1, Exponent[p, x]}]][Total[x^l]]*2^(Function[w, Sum[Product[ 2^GCD[t, l[[i]]], {i, 1, Length[l]}], {t, 1, w}]/w][If[l == {}, 1, LCM @@ l]]), {l, IntegerPartitions[n]}];

a[n_] := (b[n] - 2 b[n - 1])/4;

a /@ Range[8] (* Jean-François Alcover, Feb 19 2020, after Alois P. Heinz in A000612 *)

CROSSREFS

See A007537 for labeled case. Cf. A055621.

KEYWORD

easy,nonn

AUTHOR

Vladeta Jovovic, Jun 14 2000

EXTENSIONS

More terms from David Wasserman, Mar 21 2002

STATUS

approved

A262546

Number of Post functions of n variables which fail to satisfy Post's second condition.

+10
1

1, 1, 4, 16, 544

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

Table of n, a(n) for n=1..5.

R. F. Wheeler, Complete propositional connectives, Z. Math. Logik Grundlagen Math. 7 1961 185-198. [Annotated scanned copy]

R. F. Wheeler, An asymptotic formula for the number of complete propositional connectives, Z. Math. Logik Grundlagen Math. 8 (1962),1-4. [Annotated scanned copy]

CROSSREFS

Equals A002857 - A002543.

KEYWORD

nonn,more

AUTHOR

N. J. A. Sloane, Oct 06 2015

STATUS

approved

A262547

Nearest integer to 2^(2^n)/(4*n!).

+10
1

1, 2, 11, 683, 8947849, 6405119470038039, 16879085660760836481318184892448820, 717956902513121251386228825698709746114025202539934052824017757985572480

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Jon E. Schoenfield, Table of n, a(n) for n = 1..11

R. F. Wheeler, An asymptotic formula for the number of complete propositional connectives, Z. Math. Logik Grundlagen Math. 8 (1962), 1-4. [Annotated scanned copy]

CROSSREFS

An approximation to A002857.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Oct 06 2015

STATUS

approved

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