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A329200 consists in adding or subtracting the number A040115(n) whose digits are the differences of adjacent digits of n, depending on its parity.

Repdigits A010785 are fixed points of this map, but some numbers enter nontrivial cycles. This sequence gives the length of these cycles, ordered by their smallest member, as they are listed in the table A329196. See there for more information.

Or: Number of iterations of A329200 until a number is seen for the second time in the trajectory of n.

A329200 consists of adding to n the number whose digits are the differences of adjacent digits of n in case it is even, or subtracting it if it is odd.

The trajectory of most small numbers ends in a repdigit (A010785) which are fixed points of this map. Some larger numbers enter nontrivial cycles, cf. examples and A329196. In both cases, some number(s) will appear infinitely often in the trajectory. This sequence gives the number of iterations until a value is repeated for the first time in the trajectory of n. This is also the size of n's orbit, i.e. the total number of distinct values that will ever appear.

If n is part of the cycle, a(n) gives the length of the cycle; in particular a(n) = 1 for fixed points.

For 11 <= n <= 99 the pattern (1, 2, 5, 2, 4, 2, 3, 2, 3, 2, 3) of length 11 repeats, i.e., a(n) = a(n') if n = n' (mod 11). But the trajectory of congruent n with same a(n) does not always end in the corresponding repdigit, e.g., 11+2 and 22+2 both end in 22, 33+2 ends in 33, 44+2 ends in 44, 55+2 and 66+2 both end in 66, 77+2 and 88+2 in 77, etc.

A329200 consists of adding the number whose digits are the absoute values of differences of adjacent digits of n in case it is even, or subtracting it if it is odd. Repdigits A010785 are fixed points of this map, but some numbers enter nontrivial cycles. This sequence lists these cycles, ordered by their smallest member which is always listed first. Sequence A329197 gives the row lengths.

Whenever all terms of a cycle have the same number of digits and same initial digit, then this digit can be prefixed k times to each term to obtain a different cycle of same length, for any k >= 0. (The corresponding "ghosts" A040115(n) are then the same, so the (cyclic) first differences are also the same and add again up to 0.) This is the case for rows 1, 2, 3, ... (but not row 4 or 6) of this table. Rows 5, 7 and 8 are the second members of these three families. We could call "primitive" the cycles which are not obtained from an earlier cycle by duplicating the initial digits.

Let the decimal expansion of n be abcd...efg, say. Then a(n) has decimal expansion |a-b| |b-c| |c-d| ... |e-f| |f-g|. Leading zeros in a(n) are omitted.

From M. F. Hasler, Nov 09 2019: (Start)

This sequence coincides with A080465 up to a(109) but is thereafter completely different.

Eric Angelini calls a(n) the "ghost" of the number n and considers iterations of n -> n +- a(n) depending on parity of a(n), cf. A329200 and A329201. (End)

Sequence A040115 is most naturally extended to 0 (empty sum) for single-digit arguments; that's what we use for n < 10 here. This value is subtracted from n if even, added if odd.

A040115 is zero iff the argument is a repdigit (A010785), which therefore are the fixed points of this map A329201. All small starting values reach a fixed point, but larger values may enter a nontrivial cycle (or "loop").

See the table A329342 for the list of these cycles.

This sequence gives the number of iterations of A329623 for start value n before a repeated value first appears. Unlike the "ghost iteration" of A329200 the only fixed points are the single digits 0 to 9. See A328865 for the first repeating value.

Due to A329623(n) being significantly larger than n for some values of n, the iterative sequence can grow to infinity for some n. The first value to do so is n = 1373. This appears due to the occurrence of the digit string '62637' at the end of the term of the third iteration. These five digits reappear every second iteration at the end of the term, but with more and more digits preceding it. A329917 lists other divergent n values.

The smallest value, for n >= 10, which acts as an end point before repeating is 9, which is the final value for many starting values.

The digit string '8091' forms the basis of a very long convergent series for some values of n. The digit string consisting of an arbitrary number of copies of '80' followed by '91' will eventually converge to 8091, then 891, then 91, which finally converges in ten more iterations. We can thus form a number of arbitrary length using this rule which is guaranteed to converge. This sequence appears naturally with the starting value n = 139100 which converges to 9 after 136 iterations after reaching a term with 72 digits after 20 iterations. See the linked file below.

From M. F. Hasler, Dec 03 2019: (Start)

It seems the a(n) = -1 are conjectural, i.e., we have no proof that the terms for which the trajectory seems to "explode" do not eventually end up in a cycle. For example, the 8th iterate of 1373 is 5218725017016262626273. If the 2nd digit is changed from 2 to 0, then the further iterates appear to explode up to a length of 157 bits, but finally end up in a 2-cycle of 41-digit numbers (26...26273, 62...62637).

The "repeating values" are members of cycles, listed in A328142. Only fixed points 1, ..., 9 and 4*(10^k-1)/9 + 11, k >=3, and 6 infinite families of 2-cycles are known.

(End)

The first escape value is a(1373) = -1 (without proof). - Georg Fischer, Jul 16 2020

As A040115 forms the basis of an iterative sequence leading to A329200 and A329201, this sequence forms the basis of a similar sequence A329624. As the concatenation of the digit sum can lead to a value larger than the original term we must take the absolute value of the difference to ensure subsequent terms are always positive. The largest value in the first 10000 terms is a(9991) = 171819.

A329623(n) = |n - A053392(n)|, where A053392 is the concatenation of the sums of pairs of consecutive digits.

This is the first number which appears twice for the iteration of A329623 starting with n. See A329624 for an explanation of the sequence, and for the number of iterations required before reaching this value. Terms a(9) to a(127) are all 9's, after which the sequence shows large jumps in value, e.g., a(1673) = 62626262626262626262626262637.

All -1 are so far conjectural, see A329624 and A329917 for more information.

The terms > 0 of this sequence are elements of cycles for A329623. Only 2-cycles and fixed points 1, 2, ..., 9 and 4...455 are known. Therefore a(n) is the earliest A329623^{k}(n) = A329623^{k+2}(n) if such k exist. See A328142 for the list of all possible values and more precise definitions. - M. F. Hasler, Dec 06 2019

The first escape value is a(1373) = -1. - Georg Fischer, Jul 16 2020

A329201 consists of adding or subtracting the number whose digits are the differences of adjacent digits of n, depending on its parity. Repdigits A010785 are fixed points of this map, but some numbers enter nontrivial cycles.

This sequence lists these cycles, ordered by their smallest member which is always listed first.

Sequence A329341 gives the lengths of these cycles, i.e., rows of this table.

Whenever all terms of a cycle have the same number of digits and same initial digit, then this digit can be prefixed k times to each term to obtain a different cycle of same length, for any k >= 0. (The corresponding "ghosts" A040115(n) are then the same, so the first differences are also the same and add again up to 0.) This is the case for rows 3, 4, 5, 6, ... of this table. Rows 7, 8, 11, ... are subsequent members of the respective family. We could call "primitive" the cycles which are not obtained from an earlier cycle by duplicating the initial digits.

These are the starting values n for which the trajectory under iterations of A329623 grows without limit. See A329624 for an explanation of the sequence.

There are 466 terms below 10000.

From M. F. Hasler, Dec 02 2019: (Start)

There is no term below 10^3, but beyond 10^4 they become much more frequent: roughly 1/3 of all numbers between 10^4 and 5*10^4 are in the sequence.

The sequence consists mostly in runs of consecutive numbers ending with the next larger term with final digit 9: 1373..1379, 1382..1389, 1391..1399, 1591..1599, 1891..1899, 2373..2379, ... In some ranges, like a(152) = 4010 to a(240) = 4181, a(307) = 6010 to a(352) = 6391, a(467) = 10010 to a(556) = 11090, ..., this pattern is reversed: the runs start with a multiple of 10.

However, all these terms are so far only conjectural. We have no proof that the terms for which the trajectory seems to "explode" do not eventually end up in a cycle. For example, the 8th iterate of a(1) = 1373 is 5218725017016262626273. If the 2nd digit is changed from 2 to 0, then the further iterates grow to a length of 52 digits, but finally end up in a 2-cycle of 45-digit numbers (26...26273, 62...62637). (All members of cycles are listed in A328142.)

(End)