This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A007062 M0966 #48 Apr 24 2024 08:21:12
%S A007062 1,2,4,5,7,12,14,15,23,28,30,41,43,48,56,67,69,84,86,87,111,116,124,
%T A007062 139,141,162,180,181,183,224,232,237,271,276,278,315,333,338,372,383,
%U A007062 385,426,428,439,473,478,538,543,551,556,620
%N A007062 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 A007062 From _Gerald McGarvey_, Aug 05 2004: (Start)
%C A007062 Consider the following array:
%C A007062 .1..2..3..4..5..6..7..8..9.10.11.12.13.14.15.16.17.18.19.20
%C A007062 .2..1..4..3..6..5..8..7.10..9.12.11.14.13.16.15.18.17.20.19
%C A007062 .4..1..2..5..6..3.10..7..8.11.12..9.16.13.14.17.18.15.22.19
%C A007062 .5..2..1..4..7.10..3..6..9.12.11..8.17.14.13.16.19.22.15.18
%C A007062 .7..4..1..2..5.12..9..6..3.10.13.14.17..8.11.18.15.22.19.16
%C A007062 12..5..2..1..4..7.14.13.10..3..6..8.22.15.18.11..8.17.24.23
%C A007062 14..7..4..1..2..5.12.15.22..8..6..3.10.13.20.23.24.17..8.11
%C A007062 15.12..5..2..1..4..7.14.23.20.13.10..3..6..8.22.25.28.31.18
%C A007062 23.14..7..4..1..2..5.12.15.28.25.22..8..6..3.10.13.20.33.30
%C A007062 28.15.12..5..2..1..4..7.14.23.30.33.20.13.10..3..6..8.22.25
%C A007062 which is formed as follows:
%C A007062 . first row is the positive integers
%C A007062 . 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 A007062 . n-th row: group the (n-1)st row in groups of n and reverse the order within groups
%C A007062 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 A007062 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 that lim_{n->oo} a(n)/n^2 exists and is ~ 0.22847 ~ sqrt(0.0522). (End)
%D A007062 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007062 Peter J. C. Moses, <a href="/A007062/b007062.txt">Table of n, a(n) for n = 1..5000</a>
%H A007062 Clark Kimberling and David Callan, <a href="http://www.jstor.org/stable/2323261">Problem E3163</a>, Amer. Math. Monthly, 96 (1989), 57.
%F A007062 Conjecture: a(n) = A057030(n-1) + 1 for n > 1 with a(1) = 1. - _Mikhail Kurkov_, Feb 24 2023
%e A007062 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 A007062 (* works per the name description *)
%t A007062 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 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 *)
%Y A007062 Cf. A057030 (here we have "s(1), ..., s(n)", whereas 057030 has "s(1), ..., s(n-1)").
%Y A007062 Cf. A057032, A057033, A057063, A057064.
%K A007062 nonn
%O A007062 1,2
%A A007062 _N. J. A. Sloane_, _Mira Bernstein_
%E A007062 More terms and better description from _Clark Kimberling_, Jul 28 2000