A321011 Trajectory of 86 under repeated application of the map k -> A320486(k^2).

86, 7396, 547816, 12985, 805, 648025, 1325, 1762, 3106, 94723, 86, 7396, 547816, 12985, 805, 648025, 1325, 1762, 3106, 94723, 86, 7396, 547816, 12985, 805, 648025, 1325, 1762, 3106, 94723, 86, 7396, 547816, 12985, 805, 648025, 1325, 1762, 3106, 94723

Offset: 1

Author

N. J. A. Sloane, Nov 04 2018

Keywords

• nonn
• base
• easy

Comments

k -> A320486(k) is Eric Angelini's remove-repeated-digits map.

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:

- reach 0,

- reach one of the four fixed points 1, 1465, 4376, 89476 (see A321010)

- reach the period-10 cycle shown in A321011, or

- reach the period-9 cycle shown in A321012.

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).

Examples

			The cycle of length 10 is (86, 7396, 547816, 12985, 805, 648025, 1325, 1762, 3106, 94723).
		

References

• Eric Angelini, Postings to Sequence Fans Mailing List, Oct 24 2018 and Oct 26 2018.

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,1).

Crossrefs

Cf. A320485, A320486, A321010, A321012.

Programs

Mathematica

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 *)

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

Formula

From Colin Barker, Nov 04 2018: (Start)

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)).

a(n) = a(n-10) for n>10.

(End)