A350311 -id:A350311 - cp's OEIS frontend

A381933 a(n) is the number of occurrences of n in A350311.

1, 1, 1, 2, 2, 2, 3, 3, 2, 4, 4, 3, 4, 5, 4, 5, 6, 4, 5, 7, 6, 5, 8, 7, 6, 8, 7, 5, 9, 9, 6, 9, 10, 8, 10, 11, 7, 10, 11, 8, 8, 12, 10, 10, 13, 10, 10, 14, 13, 9, 15, 14, 11, 14, 14, 10, 14, 15, 9, 12, 16, 13, 13, 18, 14, 14, 18, 15, 11, 19, 18, 13, 18, 19, 15

Offset: 0

Rémy Sigrist, Mar 10 2025

			The value 7 appears three times in A350311, hence a(7) = 3.

Rémy Sigrist, Table of n, a(n) for n = 0..8191
Rémy Sigrist, PARI program

Cf. A350311.

PARI
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A350215 A048715, written in binary.

Original entry on oeis.org

0, 1, 10, 100, 1000, 1001, 10000, 10001, 10010, 100000, 100001, 100010, 100100, 1000000, 1000001, 1000010, 1000100, 1001000, 1001001, 10000000, 10000001, 10000010, 10000100, 10001000, 10001001, 10010000, 10010001, 10010010, 100000000, 100000001, 100000010

Offset: 0

A.H.M. Smeets, Dec 19 2021

Narayana weighted representation of n (the top version).

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

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

Cf. A000930, A048715, A350311, A350312.

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

bin[n_] := FromDigits[IntegerDigits[n, 2]]; q[n_] := BitAnd[n, 6*n] == 0; bin /@ Select[Range[0, 250], q] (* Amiram Eldar, Jan 27 2022 *)

def c(b): return not "11" in b and not "101" in b
def auptod(digits):
    return [int(b) for b in (bin(k)[2:] for k in range(2**digits)) if c(b)]
print(auptod(9)) # Michael S. Branicky, Dec 20 2021

Regular expression 0|(1000*)*10*.

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

Original entry on oeis.org

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

A.H.M. Smeets, Dec 24 2021

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

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

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

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

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 *)

# 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

# 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

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

cp's OEIS Frontend

A381933 a(n) is the number of occurrences of n in A350311.

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A350215 A048715, written in binary.

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Python

Formula