A321011 -id:A321011 - cp's OEIS frontend

A321012 Trajectory of 596 under repeated application of the map k -> A320486(k^2).

596, 3216, 103425, 197325, 897162, 652, 2510, 631, 3986, 596, 3216, 103425, 197325, 897162, 652, 2510, 631, 3986, 596, 3216, 103425, 197325, 897162, 652, 2510, 631, 3986, 596, 3216, 103425, 197325, 897162, 652, 2510, 631, 3986, 596, 3216, 103425, 197325

Offset: 1

N. J. A. Sloane, Nov 04 2018

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).

			The cycle of length 9 is (596, 3216, 103425, 197325, 897162, 652, 2510, 631, 3986).

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

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

Cf. A320485, A320486, A321010, A321011.

PadRight[{},80,{596,3216,103425,197325,897162,652,2510,631,3986}] (* Harvey P. Dale, Aug 08 2023 *)

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

From Colin Barker, Nov 04 2018: (Start)
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)).
a(n) = a(n-9) for n>9.
(End)

A321010 Numbers k such that f(k^2) = k, where f is Eric Angelini's remove-repeated-digits map x->A320486(x).

Original entry on oeis.org

0, 1, 1465, 4376, 89476

Offset: 1

N. J. A. Sloane, Nov 03 2018

Lars Blomberg has discovered that if we start with any positive integer and repeatedly apply the map m -> A320486(m^2) then we will eventually either:
- reach 0,
- reach one of the four fixed points 1, 1465, 4376, 89476 (this sequence),
- reach the period-10 cycle shown in A321011, or
- reach the period-9 cycle shown in A321012.
From Lars Blomberg, Nov 17 2018: (Start)
Verified by testing all possible 8877690 start values that these are the only fixed points and cycles.
Detailed counts are:
- 561354 reach 0,
- 963738 reach one of the four fixed points 1, 1465, 4376, 89476 (counts 946109, 434, 17065, 130),
- 7271337 reach the period-10 cycle, and
- 81261 reach the period-9 cycle. (End)

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

N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides. (Mentions this sequence)

Cf. A320485, A320486, A321011, A321012.

cp's OEIS Frontend

A321012 Trajectory of 596 under repeated application of the map k -> A320486(k^2).

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