A383423 Indices k such that A003849(k) = 0 and A383422(k) = 0.

0, 3, 7, 10, 11, 15, 18, 21, 28, 29, 32, 36, 39, 44, 47, 50, 54, 57, 58, 62, 65, 68, 75, 76, 79, 83, 86, 87, 91, 94, 97, 104, 105, 109, 112, 115, 120, 123, 126, 130, 133, 134, 138, 141, 144, 151, 152, 155, 159, 162, 167, 170, 173, 180, 181, 185, 188, 191

Offset: 1

Author

Clark Kimberling, Apr 27 2025

Keywords

• nonn

Comments

The positive integers are partitioned by this sequence together with A383424, A383425, and A383426.

Conjecture: {a(n) - a(n-1), n>=2} = {1, 3, 4, 5, 7}.

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A003849, A383422, A383424, A383425, A383426, A383427.

Programs

Mathematica

wF = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0}}] &, {0}, 15];  (* A003849 *)
s[0] = "0"; s[1] = "011"; s[n_] := StringJoin[s[n - 1], s[n - 2]]; (* A383422 *)
wL = Join[{0}, IntegerDigits[FromDigits[s[15]]]];
-1+Select[Range[400], wF[[#]] == wL[[#]] == 0 &]

Python

from itertools import count, islice
from math import isqrt
def A383423_gen(): # generator of terms
    for n in count(1):
        if ((k:=(n+isqrt(5*n**2)&-2)-n)+1+isqrt(m:=5*(k+1)**2)>>1)-(k+isqrt(m-10*k-5)>>1)==2: yield k-1
A383423_list = list(islice(A383423_gen(),58)) # Chai Wah Wu, May 22 2025