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%I M0966 #41 Apr 21 2024 21:07:04
%S 1,2,4,5,7,12,14,15,23,28,30,41,43,48,56,67,69,84,86,87,111,116,124,
%T 139,141,162,180,181,183,224,232,237,271,276,278,315,333,338,372,383,
%U 385,426,428,439,473,478,538,543,551,556,620
%N 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).
%C From _Gerald McGarvey_, Aug 05 2004: (Start)
%C Consider the following array:
%C .1..2..3..4..5..6..7..8..9.10.11.12.13.14.15.16.17.18.19.20
%C .2..1..4..3..6..5..8..7.10..9.12.11.14.13.16.15.18.17.20.19
%C .4..1..2..5..6..3.10..7..8.11.12..9.16.13.14.17.18.15.22.19
%C .5..2..1..4..7.10..3..6..9.12.11..8.17.14.13.16.19.22.15.18
%C .7..4..1..2..5.12..9..6..3.10.13.14.17..8.11.18.15.22.19.16
%C 12..5..2..1..4..7.14.13.10..3..6..8.22.15.18.11..8.17.24.23
%C 14..7..4..1..2..5.12.15.22..8..6..3.10.13.20.23.24.17..8.11
%C 15.12..5..2..1..4..7.14.23.20.13.10..3..6..8.22.25.28.31.18
%C 23.14..7..4..1..2..5.12.15.28.25.22..8..6..3.10.13.20.33.30
%C 28.15.12..5..2..1..4..7.14.23.30.33.20.13.10..3..6..8.22.25
%C which is formed as follows:
%C . first row is the positive integers
%C . 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
%C . n-th row: group the (n-1)st row in groups of n and reverse the order within groups
%C 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'.
%C 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)
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Peter J. C. Moses, <a href="/A007062/b007062.txt">Table of n, a(n) for n = 1..5000</a>
%H Clark Kimberling and David Callan, <a href="http://www.jstor.org/stable/2323261">Problem E3163</a>, Amer. Math. Monthly, 96 (1989), 57.
%F a(n) = A057030(n-1) + 1 for n > 1 with a(1) = 1. - _Mikhail Kurkov_, Feb 24 2023 [verification needed]
%e 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.
%t (*works per the name description*)
%t 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
%t (*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 *)
%o (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 [verification needed]
%Y Cf. A057030 (here we have "s(1), ..., s(n)", whereas 057030 has "s(1), ..., s(n-1)").
%Y Cf. A057032, A057033, A057063, A057064.
%K nonn,changed
%O 1,2
%A _N. J. A. Sloane_, _Mira Bernstein_
%E More terms and better description from _Clark Kimberling_, Jul 28 2000
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