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A226251
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Concatenated cyclical sequence starting from Fibonacci sequence.
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0
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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, 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, 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
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OFFSET
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1,3
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LINKS
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FORMULA
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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.
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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