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%I #16 Aug 08 2023 10:22:39
%S 596,3216,103425,197325,897162,652,2510,631,3986,596,3216,103425,
%T 197325,897162,652,2510,631,3986,596,3216,103425,197325,897162,652,
%U 2510,631,3986,596,3216,103425,197325,897162,652,2510,631,3986,596,3216,103425,197325
%N Trajectory of 596 under repeated application of the map k -> A320486(k^2).
%C k -> A320486(k) is Eric Angelini's remove-repeated-digits map.
%C _Lars Blomberg_ has discovered that if we start with any positive integer and repeatedly apply the map k -> A320486(k^2) then we will eventually either:
%C - reach 0,
%C - reach one of the four fixed points 1, 1465, 4376, 89476 (see A321010)
%C - reach the period-10 cycle shown in A321011, or
%C - reach the period-9 cycle shown in A321012.
%C Since there are only finitely many possible starting values with all digits distinct, it should not be difficult to check that this is true (and indeed, _Lars Blomberg_ may by now have completed the proof).
%D Eric Angelini, Postings to Sequence Fans Mailing List, Oct 24 2018 and Oct 26 2018.
%H Colin Barker, <a href="/A321012/b321012.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,1).
%F From _Colin Barker_, Nov 04 2018: (Start)
%F G.f.: x*(596 + 3216*x + 103425*x^2 + 197325*x^3 + 897162*x^4 + 652*x^5 + 2510*x^6 + 631*x^7 + 3986*x^8) / ((1 - x)*(1 + x + x^2)*(1 + x^3 + x^6)).
%F a(n) = a(n-9) for n>9.
%F (End)
%e The cycle of length 9 is (596, 3216, 103425, 197325, 897162, 652, 2510, 631, 3986).
%t PadRight[{},80,{596,3216,103425,197325,897162,652,2510,631,3986}] (* _Harvey P. Dale_, Aug 08 2023 *)
%o (PARI) Vec(x*(596 + 3216*x + 103425*x^2 + 197325*x^3 + 897162*x^4 + 652*x^5 + 2510*x^6 + 631*x^7 + 3986*x^8) / ((1 - x)*(1 + x + x^2)*(1 + x^3 + x^6)) + O(x^40)) \\ _Colin Barker_, Nov 04 2018
%Y Cf. A320485, A320486, A321010, A321011.
%K nonn,base,easy
%O 1,1
%A _N. J. A. Sloane_, Nov 04 2018
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