(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 word-sequence 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 |
| '--------------------'
