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A007062
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Let P(n) of a sequence s(1),s(2),s(3),... be obtained by leaving s(1),...,s(n) fixed and reversing every n consecutive terms thereafter; apply P(2) to 1,2,3,... to get PS(2), then apply P(3) to PS(2) to get PS(3), then apply P(4) to PS(3), etc. This sequence is the limit of PS(n).
(Formerly M0966)
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12
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1, 2, 4, 5, 7, 12, 14, 15, 23, 28, 30, 41, 43, 48, 56, 67, 69, 84, 86, 87, 111, 116, 124, 139, 141, 162, 180, 181, 183, 224, 232, 237, 271, 276, 278, 315, 333, 338, 372, 383, 385, 426, 428, 439, 473, 478, 538, 543, 551, 556, 620
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OFFSET
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1,2
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COMMENTS
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Consider the following array:
.1..2..3..4..5..6..7..8..9.10.11.12.13.14.15.16.17.18.19.20
.2..1..4..3..6..5..8..7.10..9.12.11.14.13.16.15.18.17.20.19
.4..1..2..5..6..3.10..7..8.11.12..9.16.13.14.17.18.15.22.19
.5..2..1..4..7.10..3..6..9.12.11..8.17.14.13.16.19.22.15.18
.7..4..1..2..5.12..9..6..3.10.13.14.17..8.11.18.15.22.19.16
12..5..2..1..4..7.14.13.10..3..6..8.22.15.18.11..8.17.24.23
14..7..4..1..2..5.12.15.22..8..6..3.10.13.20.23.24.17..8.11
15.12..5..2..1..4..7.14.23.20.13.10..3..6..8.22.25.28.31.18
23.14..7..4..1..2..5.12.15.28.25.22..8..6..3.10.13.20.33.30
28.15.12..5..2..1..4..7.14.23.30.33.20.13.10..3..6..8.22.25
which is formed as follows:
. first row is the positive integers
. second row: group the first row in pairs of two and reverse the order within groups; e.g., 1 2 -> 2 1 and 3 4 -> 4 3
. n-th row: group the (n-1)st row in groups of n and reverse the order within groups
This sequence is the first column of this array, as well as the diagonal excluding the diagonal's first term. It is also various other 'partial columns' and 'partial diagonals'.
To calculate the i-th column / j-th row value, one can work backwards to find which column of the first row it came from. For each row, first reverse its position within the group then go up. It appears lim_{n->infinity} a(n)/n^2 exists and is ~ 0.22847 ~ sqrt(0.0522) (end)
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Clark Kimberling and David Callan, Problem E3163, Amer. Math. Monthly, 96 (1989), 57.
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FORMULA
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EXAMPLE
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PS(2) begins with 1,2,4,3,6,5,8; PS(3) with 1,2,4,5,6,3,10; PS(4) with 1,2,4,5,7,10,3.
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MATHEMATICA
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(*works per the name description*)
a007062=Range[x=3500]; Do[a007062=Flatten[Join[{Take[a007062, n]}, Map[Reverse, Partition[Drop[a007062, n], n]]]], {n, 2, NestWhile[#+1&, 1, (x=# Floor[x/#])>0&]-1}]; a007062
(*works by making McGarvey's array*)a=Range[x=10000]; rows=Table[a=Flatten[Map[Reverse, Partition[a, n]]], {n, NestWhile[#+1&, 1, (x=# Floor[x/#])>0&]-1}]; a007062=Map[First, rows] (* Peter J. C. Moses, Nov 10 2016 *)
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PROG
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(PARI) a(n) = if(n==1, 1, my(n=n-1, A=n, B); for(i=1, n-1, B=n-i; A=A+B-2*(A%B)); A+1) \\ Mikhail Kurkov, Feb 24 2023
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CROSSREFS
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Cf. A057030 (here we have "s(1), ..., s(n)", whereas 057030 has "s(1), ..., s(n-1)").
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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