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

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

Author

A.H.M. Smeets, Dec 24 2021

Keywords

• nonn
• base

Comments

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

Links

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

Crossrefs

Cf. A000930, A048715, A350215, A350312.

Cf. A022290 (analog for Fibonacci numbers).

Programs

Maple

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

PARI

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

Python

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