OFFSET
1,3
FORMULA
Start with the Fibonacci sequence 0,1,1,2,3,5,8; at this point rewrite the next term 13 as 1,3 and continue adding: 1,3,4,7. At this point rewrite the sum 11 as 1,1 and the sequence will recur if values greater than or equal to 10 are rewritten as two single-digit values as 0,1,1,2,3,5,8,1,3,4,7,1,1,2,3,5,8,1,3,4,7,1,1,...
Another interesting note: The sequence can also be generated from any two numbers that do not sum to fourteen.
See Examples below.
EXAMPLE
1) (9,9) 9,9,1,8,9,1,7,8,1,5,6,1,1,2,3,5,8,1,3,4,7,1, 1, 2, 3, 5, 8, 1, 3, 4, 7,1, 1, 2, 3, 5, 8, 1, 3, 4, 7, ...
2) (3,7) 3,7,9,1,6,7,1,3,4,7,1,1,2,3,5,8,1,3,4,7,1, 1, 2, 3, 5, 8, 1, 3, 4, 7,1, 1, 2, 3, 5, 8, 1, 3, 4, 7, ...
3) (0,4) 0,4,4,8,1,2,3,5,8,1,3,4,7,1,1,2,3,5,8,1,3,4,7,1, 1, 2, 3, 5, 8, 1, 3, 4, 7,1, 1, 2, 3, 5, 8, 1, 3, 4, 7, ...
However if the sequence starts with one of the following,
(0,7),(1,4),(2,6),(3,1),(4,2),(4,5),(5,9),(6,8),(7,0),(7,7),(8,6),(9,5) the sequence converges to 1,4,5,9 which is listed as a subsequence of A000285. For all but the trivial exception (0,0) the rest of the two-digit combinations when added together will generate the sequence.
MATHEMATICA
nxt[{a_, b_}]:=If[b>9, IntegerDigits[b], {b, a+b}]; NestList[nxt, {1, 1}, 120][[All, 1]] (* or *) PadRight[{}, 120, {1, 1, 2, 3, 5, 8, 1, 3, 4, 7}] (* Harvey P. Dale, Jan 07 2020 *)
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Lynn R. Purser, Jun 03 2013
EXTENSIONS
More terms from Harvey P. Dale, Jan 07 2020
STATUS
approved