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

1,1

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).

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).

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)

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A320485, A320486, A321010, A321012.

Sequence in context: A223828 A223786 A223972 * A093286 A093243 A223814

Adjacent sequences: A321008 A321009 A321010 * A321012 A321013 A321014

Keyword

nonn,base,easy

Author

N. J. A. Sloane, Nov 04 2018

Status

approved