# Twist-onacci Sequence

I started with a well known sequence that follows a simple rule:

1, 1, 2, 3, 5, 8, 13, 21, 34...

Then I decided to twist it up a bit, and got this sequence:

1, 1, 4, 5, 8, 10, 15, 17, 23...

I thought that was interesting, so I changed the rules a different way and got this:

1, 1, 4, 7, 11, 16, 22, 28, 34...

Hoping that combining both changes would make something even more interesting, I was disappointed that it generated this:

1, 1, 6, 6, 6, 6, 6, 6, 6...

First, can you tell me how I generated each sequence. Once you've figured that out, see if you can change the starting values of the last sequence so it doesn't end in a loop.

Good luck!

Edit: Upon closer examination, the second task may not be possible. So instead, I would like a proof (formal or informal) that all possible inputs will eventually end in a loop.

(Now with correct handling of numbers above 9, plus the longest loop.)

Explanations for the sequences

These were clued in by familiarity with sequence A and by the tag.

      Sequence A
1,    1,    2,    3,    5,    8,    13,   21,   34, . . .


    |     |     |     |     |     |      |     |     |
1  +  1  =  2     |     |     |      |     |     |
1  +  2  =  3     |     |      |     |     |
2  +  3  =  5     |      |     |     |
3  +  5  =  8      |     |     |
5  +  8  =  13     |     |
.     .     |  

      Sequence B
1,     1,      4,      5,       8,        10,   15, 17, 23, . . .


    |      |       |       |        |          |     |   |   |
1 + "one"  =   4       |        |          |     |   |   |      "one" = 3 (letters)
1 + "four"  =   5        |          |     |   |   |     "four" = 4
4 + "five"  =    8          |     |   |   |     "five" = 4
5 + "eight"  =     10     |   |   |    "eight" = 5
8 + "onezero" = 15   |   |  "onezero" = 7
.     .   .   |  

      Sequence C
1,    1,     4,      7,      11,      16,   22,  28,  34, . . .


    |     |      |       |        |        |     |    |    |
"one" +  1  =   4       |        |        |     |    |    |
"one" +   4  =    7        |        |     |    |    |     "one" = 3 (letters)
"four" +    7  =    11        |     |    |    |    "four" = 4
"seven" +    11  =    16     |    |    |   "seven" = 5
"oneone" +    16  = 22    |    |  "oneone" = 6
"onesix" + 22 = 28    |  "onesix" = 6
.    .    .  

      Sequence D
1,      1,      6,      6,      6,   6,   6,   6,   6, . . .


    |       |       |       |       |    |    |    |    |
"one" + "one" =    6       |       |    |    |    |    |  "one" = 3 (letters)
"one" + "six" =    6       |    |    |    |    |  "six" = 3
"six" + "six" =    6    |    |    |    |
"six" + "six" = 6    |    |    |
.    .    .    |    |
.    .    .    |  

Informal proof that a sequence like D is destined to loop

All numbers above 14 (“onefour”) are more than twice the letter counts of their spellings.

   1  <  2 x  3 "one"         8  <  2 x  5 "eight"         15  >  2 x  7 "onefive"
2  <  2 x  3 "two"         9   > 2 x  4 "nine"          16  >  2 x  6 "onesix"
3  <  2 x  5 "three"      10  <  2 x  7 "onezero"       17  >  2 x  8 "oneseven"
4  <  2 x  4 "four"       11  <  2 x  6 "oneone"        18  >  2 x  8 "oneeight"
5  <  2 x  4 "five"       12  =  2 x  6 "onetwo"        19  >  2 x  7 "onenine"
6  =  2 x  3 "six"        13  <  2 x  8 "onethree"      20  >  2 x  7 "twozero"
7  <  2 x  5 "seven"      14  =  2 x  7 "onefour"       21  >  2 x  6 "twoone"  
⇒ The sum of letter counts in the spellings of any two numbers above 14 will be less than the larger of those two numbers.

⇒ The sequence cannot increase indefinitely.

⇒ The sequence will have a limited set of possibilities for consecutive-term pairs.

⇒ At least one consecutive-term pair will be repeated.

⇒ A repeated consecutive-term pair will be followed by a loop because each successive term is determined solely by its preceding consecutive-term pair.

Longest loop for a sequence like D

     Sequence E
9,       9,        8,       9,       9,        8,   9,   9,   8,   9, . . .

"nine" + "nine" =      8        |        |         |    |    |    |    |   "nine" = 4
"nine" + "eight" =     9        |         |    |    |    |    |  "eight" = 5
"eight" + "nine" =     9         |    |    |    |    |
"nine" + "nine" =      8    |    |    |    |
"nine" + "eight" = 9    |    |    |
.    .    .    |    |
.    .    .    |  

Sequence E, with a repeating cycle of 3 terms, seems to be the only loop that resembles sequence D that also repeats more than one term. Three loops repeat single terms: 6 (sequence D itself), 12 or 14.

Sequence E was found by looking at sequence D’s addition table, but only the portion with numbers that can be sums as well as summands.

                                     oneone    onezero
seven     onetwo    onefour
six      nine     eight     onesix    onefive   onethree
+       3        4         5         6          7         8
____________________________________________________________
| .-----.                                                    .-------------------.
3  | | six |   seven     eight      nine     onezero    oneone  | Loops may be      |
| |  3  |     5         5         4          7         6     | found by starting |
| '-----'            .------.                                | from cells within |
4  |  seven    eight    | nine |  onezero     oneone    onetwo  | the outlined      |
|    5        5      |  4   |     7          6         6     | regions, which    |
|          .------.  '------'                                | contain every sum |
5  |  eight   | nine |  onezero    oneone     onetwo   onethree | that is no larger |
|    5     |  4   |     7         6          6         8     | than either of    |
|          '------'           .--------.                     | its summands.     |
6  |  nine    onezero    oneone  | onetwo |  onethree  onefour  |                   |
|    4        7         6     |   6    |     8         7     | Any loop includes |
|                             '--------'.--------------------' at least one such |
7  | onezero   oneone    onetwo   onethree | onefour   onefive    sum.  (Though not |
|    7        6         6         8     |    7         7       all such sums are |
|                                       |                      parts of loops.)  |
8  |  oneone   onetwo   onethree  onefour  | onefive    onesix  .-------------------'
|    6        6         8         7     |    7         6     |
|                                       '--------------------'  

• That's it! I debated putting the "word-sequence" tag in, because I thought it might make it too easy, but decided to keep it anyways. Commented Oct 6, 2017 at 21:35
• Relevant(ish) video about these sequences Commented Oct 7, 2017 at 10:15
• Quite a relevant video, @Holloway, motivating the addition of what seems to be the longest cycle possible for the last sequence here
– humn
Commented Oct 8, 2017 at 6:11
• Thanks for your awesome ASCII art. If you used Imgur, I would be in trouble (see my profile). Commented Jun 25, 2020 at 2:03