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%I #18 Nov 05 2020 13:44:59
%S 86,7396,547816,12985,805,648025,1325,1762,3106,94723,86,7396,547816,
%T 12985,805,648025,1325,1762,3106,94723,86,7396,547816,12985,805,
%U 648025,1325,1762,3106,94723,86,7396,547816,12985,805,648025,1325,1762,3106,94723
%N Trajectory of 86 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="/A321011/b321011.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,0,1).
%F From _Colin Barker_, Nov 04 2018: (Start)
%F G.f.: x*(86 + 7396*x + 547816*x^2 + 12985*x^3 + 805*x^4 + 648025*x^5 + 1325*x^6 + 1762*x^7 + 3106*x^8 + 94723*x^9) / ((1 - x)*(1 + x)*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)).
%F a(n) = a(n-10) for n>10.
%F (End)
%e The cycle of length 10 is (86, 7396, 547816, 12985, 805, 648025, 1325, 1762, 3106, 94723).
%t LinearRecurrence[{0,0,0,0,0,0,0,0,0,1},{86,7396,547816,12985,805,648025,1325,1762,3106,94723},40] (* or *) PadRight[ {},40,{86,7396,547816,12985,805,648025,1325,1762,3106,94723}] (* _Harvey P. Dale_, Nov 05 2020 *)
%o (PARI) Vec(x*(86 + 7396*x + 547816*x^2 + 12985*x^3 + 805*x^4 + 648025*x^5 + 1325*x^6 + 1762*x^7 + 3106*x^8 + 94723*x^9) / ((1 - x)*(1 + x)*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)) + O(x^40)) \\ _Colin Barker_, Nov 04 2018
%Y Cf. A320485, A320486, A321010, A321012.
%K nonn,base,easy
%O 1,1
%A _N. J. A. Sloane_, Nov 04 2018
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