A350312 -id:A350312 - cp's OEIS frontend

A350215 A048715, written in binary.

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*.

A350311 Replace 2^k in the binary expansion of n with A000930(k+2), Narayana's cows sequence.

Original entry on oeis.org

0, 1, 2, 3, 3, 4, 5, 6, 4, 5, 6, 7, 7, 8, 9, 10, 6, 7, 8, 9, 9, 10, 11, 12, 10, 11, 12, 13, 13, 14, 15, 16, 9, 10, 11, 12, 12, 13, 14, 15, 13, 14, 15, 16, 16, 17, 18, 19, 15, 16, 17, 18, 18, 19, 20, 21, 19, 20, 21, 22, 22, 23, 24, 25, 13, 14, 15, 16, 16, 17

Offset: 0

A.H.M. Smeets, Dec 24 2021

A048715(n) = m, if and only if a(n) = m and for all k > n a(k) > m.

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

Cf. A000930, A048715, A350215, A350312.

Cf. A022290 (analog for Fibonacci numbers).

b:= (n, i, j, k)->`if`(n=0, 0, k*irem(n, 2, 'q')+b(q, j, k, i+k)):
a:= n-> b(n, 1$3):
seq(a(n), n=0..100);  # Alois P. Heinz, Jan 26 2022

my(p=Mod('x,'x^3-'x^2-1)); a(n) = vecsum(Vec(lift(subst(Pol(binary(n))*'x^2,'x,p)))); \\ Kevin Ryde, Dec 26 2021

def Interpretation(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 = 0
while n <= 69:
    print(Interpretation(n), end = ", ")
    n += 1

cp's OEIS Frontend

A350215 A048715, written in binary.

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