%I
%S 12,21,122,221,1222,2221,4615,5461,12222,22221,122222,222221,402046,
%T 640204,603069,960306,869041,904186,1222222,2222221,12222222,22222221,
%U 55887353,58873535,122222222,222222221,1222222222,2222222221,3672179309,9367217930
%N List of pairs (n,m) with n<m such that the decimal expansion of m is a cyclic shift of that of n and m^2 is a cyclic shift of n^2.
%C Inspired by David W. Wilson's messages in seqfan list.
%e 12 and 21 are rotationally connected and also their squares 144, 441 are obtained from each other by rotation of their decimal representations.
%e Also 122 and 221 are rotationally connected as well as their squares 14884 and 48841.
%e Notice infinite pattern (n,m)= (12...2, 2...21).
%e Corresponding squares:
%e {144,441},
%e {14884,48841},
%e {1493284,4932841},
%e {21298225,29822521},
%e {149377284,493772841},
%e {14938217284,49382172841},
%e {161640986116,409861161616},
%e {363692218761,922187613636},
%e {755232259681,817552322596},
%e {1493826617284,4938266172841}.
%e From _Pieter Post_, Jun 30, 2016: (Start)
%e There is another infinite subsequence:
%e The cyclic pair (201023, 320102)*k (for k = 2 and 3) and its squares (40410256529, 102465290404)*k^2.
%e The next in the sequence is:
%e (020001000203, 30200010002)*k,(400040009120406041209, 912040604120900040004)*k^2 (for k = 16, 17,..., 33).
%e In general: (0{n}20{2n+1}10{2n+1}20{n}3, 30{n}20{2n+1}10{2n+1}20{n}3)* k, where lower bound of k = 5*10^(n1)*sqr(10) and upperbound of k = 3{n+1} for n = 0, 1, 2, 3, 4, etc.
%e For example n = 6 gives lower bound k = 1581139 with lower cyclic pair:(316227800000001581139000000031622784743417, 474341731622780000000158113900000003162278)
%e and corresponding squares:(100000021492841000000214928422500007835889744785236492844223926514928486978524835889
%e ,225000078358897447852364928442239265149284869785248358891000000214928410000002149284)
%e And upperbound k = 3333333 with upper cyclic pair:(666666600000003333333000000066666669999999, 999999966666660000000333333300000006666666)
%e and its corresponding squares:
%e (444444355555564444443555555699999993333332111111222222217777778888888766666660000001,
%e 999999933333321111112222222177777788888887666666600000014444443555555644444435555556)
%e (End)
%Y Cf. A134584, A134585.
%K base,nonn,tabf
%O 1,1
%A _Zak Seidov_, Jan 12 2008
%E More terms from _Max Alekseyev_, Oct 14 2010
