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Search: id:A078389
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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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+0
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
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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a(n) = 2*a(n-1) if n is a 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 (heinz(AT)hs-heilbronn.de), Nov 23 2008]
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LINKS
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Index entries for sequences related to parenthesizing
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EXAMPLE
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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 (heinz(AT)hs-heilbronn.de), Nov 23 2008]
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MAPLE
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p:= proc (n) option remember; local x; if n<1 then {} elif n=1 then {l} 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); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Nov 23 2008]
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CROSSREFS
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Sequence in context: A056644 A007813 A005309 this_sequence A059173 A027560 A135493
Adjacent sequences: A078386 A078387 A078388 this_sequence A078390 A078391 A078392
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KEYWORD
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nonn,nice
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AUTHOR
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John W. Layman (layman(AT)math.vt.edu), May 07 2003
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EXTENSIONS
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Corrected a(5) - a(10) and extended a(11) - a(31) by Alois P. Heinz (heinz(AT)hs-heilbronn.de), Nov 23 2008
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