A359128 The Fibostracci sequence: a(0) = 0, a(1) = 1; thereafter a(n) = a(n-1)+a(n-2) if a(n-1) and a(n-2) do not share a digit, otherwise a(n) is the smallest number not yet in the sequence.

0, 1, 1, 2, 3, 5, 8, 13, 21, 4, 25, 29, 6, 35, 41, 76, 117, 7, 9, 16, 25, 41, 66, 107, 173, 10, 11, 12, 14, 15, 17, 18, 19, 20, 39, 59, 22, 81, 103, 23, 24, 26, 27, 28, 30, 58, 88, 31, 119, 32, 151, 183, 33, 34, 36, 37, 38, 40, 78, 118, 42, 160, 202, 43, 245, 44

Offset

0,4

Comments

The test is: does the set of digits in a(n-1) intersect the set of digits in a(n-2)?

Links

Michael De Vlieger, Table of n, a(n) for n = 0..16384

Eric Angelini, Fibostracci, personal blog "Cinquante Signes", blogspot.com, Sep. 30, 2022.

Eric Angelini, Fibostracci [Local copy of blog entry, with permission]

Michael De Vlieger, Log-log scatterplot of a(n), n = 1..2^20, showing a(n) > n in red else dark blue.

Example

We start the sequence (call it S) with a(1) = 0 and a(2) = 1. As 0 and 1 share no digit we add them and extend S with the sum: S = 0, 1, 1, ...
As the last two integers share at least one digit, we don't add them and extend S instead with the smallest integer not yet in S:
S = 0, 1, 1, 2, ...
As 1 and 2 share no digit, we add them and extend S with the sum:
S = 0, 1, 1, 2, 3, ...
As 2 and 3 share no digit, we add them and extend S with the sum:
S = 0, 1, 1, 2, 3, 5, ...
Then:
S = 0, 1, 1, 2, 3, 5, 8, 13, 21, ...
As the last two integers share at least one digit, we don't add them and extend S instead with the smallest integer not yet in S:
S = 0, 1, 1, 2, 3, 5, 8, 13, 21, 4, ...
Then:
S = 0, 1, 1, 2, 3, 5, 8, 13, 21, 4, 25, 29, ...
As 25 and 29 share the digit 2, we get:
S = 0, 1, 1, 2, 3, 5, 8, 13, 21, 4, 25, 29, 6, ...
And so on.

Mathematica

nn = 66; c[_] = False; Array[Set[{a[#], c[#]}, {#, True}] &, 2, 0]; i = {a[0]}; j = {a[1]}; u = 2; Do[If[IntersectingQ[i, j], k = u, k = a[n - 2] + a[n - 1]]; Set[{a[n], c[k], i, j}, {k, True, j, IntegerDigits[k]}]; If[k == u, While[c[u], u++]], {n, 2, nn}]; Array[a, nn, 0] (* Michael De Vlieger, Dec 25 2022 *)

Prog

(Python)
from itertools import islice
def agen(): # generator of terms
    aset, anm1, an, mink = {0}, 1, 0, 1
    while True:
        yield an
        anm1, an = an, mink if set(str(an)) & set(str(anm1)) else an + anm1
        aset.add(an)
        while mink in aset: mink += 1
print(list(islice(agen(), 66))) # Michael S. Branicky, Dec 25 2022

Crossrefs

Cf. A000045, A357048.

Sequence in context: A093093 A345095 A281408 * A327451 A137290 A268962

Adjacent sequences: A359125 A359126 A359127 * A359129 A359130 A359131

Keyword

nonn,base

Author

Eric Angelini, Sep 30 2022 [Submitted on his behalf by N. J. A. Sloane, Dec 25 2022]

Status

approved