A120214 -id:A120214 - cp's OEIS frontend

A118514 Define sequence S_n by: initial term = n, reverse digits and add 2 to get next term. It is conjectured that S_n always reaches a cycle. Sequence gives number of steps for S_n to reach a cycle.

1, 3, 0, 2, 0, 1, 0, 0, 0, 0, 0, 9, 0, 7, 10, 0, 9, 4, 0, 3, 8, 8, 8, 7, 15, 5, 5, 3, 12, 1, 11, 16, 0, 7, 0, 5, 8, 0, 7, 2, 0, 6, 6, 6, 6, 5, 13, 3, 3, 1, 10, 6, 9, 14, 0, 5, 0, 3, 6, 0, 5, 4, 0, 4, 4, 4, 4, 3, 11, 1, 1, 13, 8, 4, 7, 12, 0, 3, 0, 1, 4, 2, 3, 2, 0, 2, 2, 2, 2, 1, 9, 12, 0

Offset: 1

N. J. A. Sloane, May 06 2006

Initial cycles have length 81 or 90.
There is one cycle of length 81 (least component is 3, all components have at most three digits, cf. A117521), 22 cycles of length 90 with 4-digit components (least components are 1013 + 2*k for k = 0, ..., 21, cf. A120214) and one cycle of length 45 with 4-digit components (least component is 1057, cf. A120215). Furthermore there are 22 cycles of length 1890 (least components are 100013 + 2*k for k = 0, ..., 21, cf. A120216), one cycle of length 945 (least component is 100057, cf. A120217) and 225 cycles of length 900 (least components are 100103 + 2*k for k = 0, ..., 224, cf. A120218), all having 6-digit components. It is conjectured that there are also cycles of increasing length with 8-, 10-, 12-, ... digit components. - Klaus Brockhaus, Jun 10 2006
From Michael S. Branicky, May 11 2023: (Start)
There are 22 cycles of length 19890 (least components are 10000013 + 2*k for k = 0, ..., 21), one cycle of length 9945 (least component 10000057), 225 cycles of length 18900 (least components are 10000103 + 2*k for k = 0, ..., 224) and 2250 cycles of length 9000 (least components are 10001003 + 2*k for k = 0, ..., 2249), all having 8-digit components.
These patterns continue.  Specifically, there is one cycle of length 10^(n/2) - 55 (least component 10^(n-1) + 57), and there are 22 cycles of length 2*(10^(n/2) - 55) (least components 10^(n-1) + 13 + 2*k for k = 0, ..., 21), each for n = 4, 6, 8, 10, 12, 14, 16. (End)

Michael S. Branicky, Python program for OEIS A118514, A118515, and A118516
N. J. A. Sloane and others, Sequences of RADD type, OEIS wiki.

For records see A118515, A118516. Cf. A117831. S_1 is A117521.

S_1013 is A120214, S_1057 is A120215, S_100013 is A120216, S_100057 is A120217, S_100103 is A120218.

Python
```
# see linked program
```

A120216 Start with 100013 and repeatedly reverse the digits and add 2 to get the next term.

Original entry on oeis.org

100013, 310003, 300015, 510005, 500017, 710007, 700019, 910009, 900021, 120011, 110023, 320013, 310025, 520015, 510027, 720017, 710029, 920019, 910031, 130021, 120033, 330023, 320035, 530025, 520037, 730027, 720039, 930029, 920041, 140031

Offset: 1

Klaus Brockhaus, Jun 11 2006

Let T(S,Q) be the sequence obtained by starting with S and repeatedly reversing the digits and adding Q to get the next term. This is T(100013,2). 100013 is the first S for which T(S,2) reaches a cycle of length 1890. The cycle is simply the first 1890 terms, which then repeat. A full period is given in the table.

Klaus Brockhaus, Table of n, a(n) for n = 1..1890
N. J. A. Sloane and others, Sequences of RADD type, OEIS wiki.
Index entries for linear recurrences with constant coefficients, order 1890.

Cf. A117521, A118514, A120214, A120215, A120217, A120218.

NestList[FromDigits[Reverse[IntegerDigits[#]]]+2&,100013,30] (* Harvey P. Dale, Oct 03 2014 *)

A120217 Start with 100057 and repeatedly reverse the digits and add 2 to get the next term.

Original entry on oeis.org

100057, 750003, 300059, 950005, 500061, 160007, 700063, 360009, 900065, 560011, 110067, 760013, 310069, 960015, 510071, 170017, 710073, 370019, 910075, 570021, 120077, 770023, 320079, 970025, 520081, 180027, 720083, 380029, 920085, 580031

Offset: 1

Klaus Brockhaus, Jun 11 2006

Let T(S,Q) be the sequence obtained by starting with S and repeatedly reversing the digits and adding Q to get the next term. This is T(100057,2). 100057 is the first S for which T(S,2) reaches a cycle of length 945. The cycle is simply the first 945 terms, which then repeat. A full period is given in the table.

Klaus Brockhaus, Table of n, a(n) for n = 1,...,945
N. J. A. Sloane and others, Sequences of RADD type, OEIS wiki.
Index entries for linear recurrences with constant coefficients, order 945.

Cf. A117521, A118514, A120214, A120215, A120216, A120218.

A120218 Start with 100103 and repeatedly reverse the digits and add 2 to get the next term.

Original entry on oeis.org

100103, 301003, 300105, 501005, 500107, 701007, 700109, 901009, 900111, 111011, 110113, 311013, 310115, 511015, 510117, 711017, 710119, 911019, 910121, 121021, 120123, 321023, 320125, 521025, 520127, 721027, 720129, 921029, 920131, 131031

Offset: 1

Klaus Brockhaus, Jun 11 2006

Let T(S,Q) be the sequence obtained by starting with S and repeatedly reversing the digits and adding Q to get the next term. This is T(100103,2). 100103 is the first S for which T(S,2) reaches a cycle of length 900. The cycle is simply the first 900 terms, which then repeat. A full period is given in the table.

Klaus Brockhaus, Table of n, a(n) for n = 1..900
N. J. A. Sloane and others, Sequences of RADD type, OEIS wiki.
Index entries for linear recurrences with constant coefficients, order 900.

Cf. A117521, A118514, A120214, A120215, A120216, A120217.

NestList[FromDigits[Reverse[IntegerDigits[#]]]+2&,100103,30] (* Harvey P. Dale, Aug 14 2012 *)

A120215 Start with 1057 and repeatedly reverse the digits and add 2 to get the next term.

Original entry on oeis.org

1057, 7503, 3059, 9505, 5061, 1607, 7063, 3609, 9065, 5611, 1167, 7613, 3169, 9615, 5171, 1717, 7173, 3719, 9175, 5721, 1277, 7723, 3279, 9725, 5281, 1827, 7283, 3829, 9285, 5831, 1387, 7833, 3389, 9835, 5391, 1937, 7393, 3939, 9395, 5941, 1497, 7943, 3499, 9945, 5501

Offset: 1

Klaus Brockhaus, Jun 11 2006

Let T(S,Q) be the sequence obtained by starting with S and repeatedly reversing the digits and adding Q to get the next term. This is T(1057,2). 1057 is the first S for which T(S,2) reaches a cycle of length 45. The cycle is simply the first 45 terms, which then repeat. A full period is shown.

N. J. A. Sloane and others, Sequences of RADD type, OEIS wiki.
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A117521, A118514, A120214, A120216, A120217, A120218.

NestList[IntegerReverse[#]+2&,1057,50] (* Harvey P. Dale, Jan 25 2021 *)

a(n) = a(n-45). - Boštjan Gec, Sep 21 2023

cp's OEIS Frontend

A118514 Define sequence S_n by: initial term = n, reverse digits and add 2 to get next term. It is conjectured that S_n always reaches a cycle. Sequence gives number of steps for S_n to reach a cycle.

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A120216 Start with 100013 and repeatedly reverse the digits and add 2 to get the next term.

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A120217 Start with 100057 and repeatedly reverse the digits and add 2 to get the next term.

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A120218 Start with 100103 and repeatedly reverse the digits and add 2 to get the next term.

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A120215 Start with 1057 and repeatedly reverse the digits and add 2 to get the next term.

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