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A214365 Digit-wise Fibonacci: Start with 0,1; then the next term is always the sum of the earliest two consecutive digits not yet summed so far. 3
0, 1, 1, 2, 3, 5, 8, 13, 9, 4, 12, 13, 5, 3, 3, 4, 8, 8, 6, 7, 12, 16, 14, 13, 8, 3, 3, 7, 7, 5, 5, 4, 11, 11, 6, 10, 14, 12, 10, 9, 5, 2, 2, 2, 7, 7, 1, 1, 5, 5, 3, 3, 1, 9, 14, 7, 4, 4, 9, 14, 8, 2, 6, 10, 8, 6, 4, 10, 10, 5, 11, 11, 8, 13, 10, 5, 12, 10, 8, 7, 1, 8, 14, 10, 5 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Offset chosen in analogy to the classical Fibonacci sequence. But the present sequence has no term larger than 9+9=18.

As observed by H. Havermann in reply to the original post (cf. link), a run of k consecutive 18's will yield a run of (at least) 2k-1 consecutive 9's somewhere later, which in turn will yield (at least) 2k-2 consecutive 18's. Since there are such runs of sufficient length (Z. Seidov pointed out that a(n)=9 for 78532 < n < 78598), the sequence cannot become periodic.

In what precedes, a value of k >= 3 is sufficient for infinite growth. But a run of only three 9's is also sufficient because the 2 consecutive 18's will be followed by a number >= 10, which then yields four 9's and subsequently infinitely long runs of 9's, cf Example.

From Michael S. Branicky, Dec 14 2020: (Start)

Likewise, a run of k consecutive 6's will yield (at least) k-1 consecutive 12's, then 2k-3 3's, then 2k-4 6's, leading to infinite growth for k > 4.  Five consecutive 6's first occur at a(17072).

Similarly, a run of k 8's will yield k-1 16's, 2k-3 7's, 2k-4 14's, 4k-9 5's, 4k-10 10's, 8k-21 1's, 8k-22 2's, 8k-23 2's, then 8k-24 8's, leading to infinite growth for k > 3.  Four consecutive 8's first occur at a(9606). (End)

LINKS

Lars Blomberg, Table of n, a(n) for n = 0..10000

Eric Angelini (and replies from others), Fibonaccit, posts to the SeqFan list, Feb 15 2013.

Hans Havermann, Chart of the first differences of the indices of 18 in A214365

EXAMPLE

The sequence starts in the same way as the Fibonacci sequence A000045. But after 5+8=13 follows the digit-wise continuation, viz: 8+1=9, 1+3=4, 3+9=12, ... (Due to the presence of 2-digit terms, the summed digits lag more and more behind the correspondingly computed term.)

The first run of 3 consecutive 9's occurs at a(3862)=a(3863)=a(3864)=9, which then yield a(4975)=a(4976)=18, a(4977)=14 and the first run of four 9's at 6392 <= n <= 6395. [M. F. Hasler, Feb 17 2013]

PROG

(PARI) A214365(n, show=0, d=[0, 1])={show & print1(d[1]", "d[2]); for(i=2, n, n=d[1]+d[2]; show & print1(", "n); d=concat(vecextract(d, "^1"), digits(n))); n}

(Python)

def aupto(n):

  alst, remaining = [0, 1], [0, 1]

  for i in range(2, n+1):

    an = remaining.pop(0) + remaining[0]

    alst.append(an)

    remaining.extend(list(map(int, str(an))))

  return alst    # use alst[n] for a(n)

print(aupto(89)) # Michael S. Branicky, Dec 14 2020

CROSSREFS

Cf. A093086.

Sequence in context: A105995 A214094 A254032 * A104701 A309782 A300999

Adjacent sequences:  A214362 A214363 A214364 * A214366 A214367 A214368

KEYWORD

nonn,base

AUTHOR

Eric Angelini and M. F. Hasler, Feb 16 2013

STATUS

approved

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Last modified September 19 22:06 EDT 2021. Contains 347576 sequences. (Running on oeis4.)