A093086 "Fibonacci in digits": start with a(0)=0, a(1)=1; repeatedly adjoin the digits of the sum of the next two terms.

0, 1, 1, 2, 3, 5, 8, 1, 3, 9, 4, 1, 2, 1, 3, 5, 3, 3, 4, 8, 8, 6, 7, 1, 2, 1, 6, 1, 4, 1, 3, 8, 3, 3, 7, 7, 5, 5, 4, 1, 1, 1, 1, 6, 1, 0, 1, 4, 1, 2, 1, 0, 9, 5, 2, 2, 2, 7, 7, 1, 1, 5, 5, 3, 3, 1, 9, 1, 4, 7, 4, 4, 9, 1, 4, 8, 2, 6, 1, 0, 8, 6, 4, 1, 0, 1, 0, 5, 1, 1, 1, 1, 8, 1, 3, 1, 0, 5, 1, 2, 1, 0, 8, 7, 1, 8

Offset

0,4

Comments

Formally, define strings of digits S_i as follows. S_0={0}, S_1={0,1}. For n >= 1, let S_n={t_0, t_1, ..., t_z}. Then S_{n+1} is obtained by adjoining the digits of t_{n-1}+t_n to S_n. The sequence gives the limiting string S_oo.

All digits appear infinitely often, although the sequence is not periodic.

For any d, there exists an index where at least d consecutive 9's occur and there exists an index where at least d consecutive (1, 8)'s occur. - Michael S. Branicky, Nov 07 2025

Links

Hakan Icoz, Table of n, a(n) for n = 0..20000

Example

After S_6 = {0,1,1,2,3,5,8} we have 5+8 = 13, so we get
S_7 = {0,1,1,2,3,5,8,1,3}. Then 8+1 = 9, so we get
S_8 = {0,1,1,2,3,5,8,1,3,9}. Then 1+3 = 4, so we get
S_9 = {0,1,1,2,3,5,8,1,3,9,4}, and so on.

Maple

with(linalg): A:=matrix(1, 2, [0, 1]): for n from 1 to 100 do if A[1, n]+A[1, n+1]<10 then A:=concat(A, matrix(1, 1, A[1, n]+A[1, n+1])) else A:=concat(A, matrix(1, 2, [1, A[1, n]+A[1, n+1]-10])) fi od: matrix(A); # Emeric Deutsch, May 31 2005

Mathematica

Fold[Join[#, IntegerDigits[Total[#[[#2;; #2+1]]]]] &, {0, 1}, Range[100]] (* Paolo Xausa, Aug 18 2025 *)

Prog

(Python)
from itertools import count, islice
def agen():
    alst = [0, 1]
    yield from alst
    for n in count(2):
        t = alst[n-2]+alst[n-1]
        if t > 9: alst.extend([1, t-10])
        else: alst.append(t)
        yield alst[n]
print(list(islice(agen(), 106))) # Michael S. Branicky, Nov 07 2025

Crossrefs

Cf. A093087-A093098, A105967, A102085, A214365.

Sequence in context: A008963 A031324 A226251 * A093092 A031111 A380820

Adjacent sequences: A093083 A093084 A093085 * A093087 A093088 A093089

Keyword

nonn,base,easy

Author

Bodo Zinser, Mar 20 2004

Extensions

Edited by N. J. A. Sloane, Mar 20 2010

Status

approved