A350312 Narayana weighted representation of n (the bottom version). Also binary representation of numbers not containing 00 or 010 as a substring.

0, 1, 10, 11, 101, 110, 111, 1011, 1101, 1110, 1111, 10110, 10111, 11011, 11101, 11110, 11111, 101101, 101110, 101111, 110110, 110111, 111011, 111101, 111110, 111111, 1011011, 1011101, 1011110, 1011111, 1101101, 1101110, 1101111, 1110110, 1110111, 1111011

Offset: 0

Author

A.H.M. Smeets, Dec 24 2021

Keywords

• nonn
• base

Comments

a(n) equals binary representation of m, if and only if A350311(m) = n and for all k < m, A350311(k) < n.

Links

• A.H.M. Smeets, The design of greedy number representations

Crossrefs

Cf. A000930, A048715, A350215 (top version), A350311.

Fibonacci representations: A014417 (Zeckendorf), A104326 (dual Zeckendorf).

Programs

Mathematica

q[n_] := SequenceCount[IntegerDigits[n, 2], #] & /@ {{0, 0}, {0, 1, 0}} == {0, 0}; bin[n_] := FromDigits[IntegerDigits[n, 2]]; bin /@ Select[Range[0, 120], q] (* Amiram Eldar, Jan 27 2022 *)

Python

# first method (as from definition)
def A101(n):
    f0, f1, f2, r = 1, 1, 1, 0
    while n > 0:
        if n%2 == 1:
            r = r+f0
        n, f0, f1, f2 = n//2, f0+f2, f0, f1
    return r
n, a = 0, 0
while n < 36:
    if A101(a) == n:
        print(bin(a)[2:], end = ", ")
        n += 1
    a += 1

Python

# second method (as from regular expression)
def test(n):
    s, i, n1 = bin(n)[2:], 0, 2
    while i < len(s):
        if s[i] == "0":
            if n1 < 2:
                return 0
            n1 = 0
        else:
            n1 += 1
        i += 1
    return 1
n, a = 0, 0
while n < 36:
    if test(a):
        print(bin(a)[2:], end = ", ")
        n += 1
    a += 1

Formula

Regular expression: 0|11*(0111*)*(0|01|011*)?.