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A078389
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Number of different values obtained by evaluating all different parenthesizations of 1/2/3/4/.../n.
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2
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1, 1, 2, 4, 8, 16, 32, 60, 116, 192, 384, 544, 1088, 1736, 2576, 3824, 7648, 10352, 20704, 28096, 40256, 62128, 124256, 155488, 227872, 349248, 470352, 622128, 1244256, 1499232, 2998464, 3796224, 5289920, 8048544, 10668096, 12562752, 25125504
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| a(n) = 2*a(n-1) if n is an odd prime, because (p/q)/n and p/(q/n)=(p/q)*n give exactly two different values for each of the different values p/q from the parenthesizations of 1/.../n-1 and a(n) <= 2*a(n-1) if n is not a prime. [From Alois P. Heinz, Nov 23 2008]
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LINKS
| Index entries for sequences related to parenthesizing
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EXAMPLE
| For n=4, ((1/2)/3)/4=1/24, (1/2)/(3/4)=2/3, (1/(2/3))/4=3/8, 1/((2/3)/4)=6 and 1/(2/(3/4))=3/8, giving 4 different values 1/24, 3/8, 2/3 and 6. Thus a(4) = 4.
a(5) = 2*a(4) = 2*4 = 8, because 5 is a prime; the 8 different values are: 1/120, 3/40, 2/15, 5/24, 6/5, 15/8, 10/3, 30. [From Alois P. Heinz, Nov 23 2008]
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MAPLE
| p:= proc (n) option remember; local x; if n<1 then {} elif n=1 then {1} elif n=2 then {1/2} else {seq ([x/n, x*n][], x=p (n-1))} fi end: a:= n-> nops (p(n)): seq (a(n), n=1..20); # Alois P. Heinz, Nov 23 2008
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MATHEMATICA
| p[0] = {}; p[1] = {1}; p[2] = {1/2}; p[n_] := p[n] = Union[ Flatten[ Table[ {x/n, x*n}, {x, p[n - 1]}]]]; a[n_] := Length[p[n]]; A078389 = Table[an = a[n]; Print[an]; an, {n, 1, 30}] (* From Jean-François Alcover, Jan 06 2012, after Alois P. Heinz *)
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CROSSREFS
| Sequence in context: A056644 A007813 A005309 * A059173 A027560 A135493
Adjacent sequences: A078386 A078387 A078388 * A078390 A078391 A078392
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KEYWORD
| nonn,nice
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AUTHOR
| John W. Layman (layman(AT)math.vt.edu), May 07 2003
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EXTENSIONS
| Corrected a(5)-a(10) and extended a(11)-a(31) by Alois P. Heinz (heinz(AT)hs-heilbronn.de), Nov 23 2008
a(32)-a(37) from Alois P. Heinz (heinz(AT)hs-heilbronn.de), Mar 07 2011
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