A219639 -id:A219639 - cp's OEIS frontend

A219641 a(n) = n minus (number of 1's in Zeckendorf expansion of n).

0, 0, 1, 2, 2, 4, 4, 5, 7, 7, 8, 9, 9, 12, 12, 13, 14, 14, 16, 16, 17, 20, 20, 21, 22, 22, 24, 24, 25, 27, 27, 28, 29, 29, 33, 33, 34, 35, 35, 37, 37, 38, 40, 40, 41, 42, 42, 45, 45, 46, 47, 47, 49, 49, 50, 54, 54, 55, 56, 56, 58, 58, 59, 61, 61, 62, 63, 63, 66

Offset: 0

Antti Karttunen, Nov 24 2012

nonn

See A014417 for the Fibonacci number system representation, also known as Zeckendorf expansion.

Antti Karttunen, Table of n, a(n) for n = 0..10000
Paul Baird-Smith, Alyssa Epstein, Kristen Flint, and Steven J. Miller, The Zeckendorf Game, arXiv:1809.04881 [math.NT], 2018.

Cf. A007895, A014417. A022342 gives the positions of records, resulting the same sequence with duplicates removed: A219640. A035336 gives the positions of values that occur only once: A219639. Cf. also A219637, A219642. Analogous sequence for binary system: A011371, for factorial number system: A219651.

zeck = DigitCount[Select[Range[0, 500], BitAnd[#, 2*#] == 0&], 2, 1];
Range[0, Length[zeck]-1] - zeck (* Jean-François Alcover, Jan 25 2018 *)

from sympy import fibonacci
def a(n):
    k=0
    x=0
    while n>0:
        k=0
        while fibonacci(k)<=n: k+=1
        x+=10**(k - 3)
        n-=fibonacci(k - 1)
    return str(x).count("1")
print([n - a(n) for n in range(101)]) # Indranil Ghosh, Jun 09 2017

Scheme
```
(define (A219641 n) (- n (A007895 n)))
```

a(n) = n - A007895(n).

A219640 Numbers m for which there exists k such that m = k - (number of 1's in Zeckendorf expansion of k); distinct values in A219641.

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 8, 9, 12, 13, 14, 16, 17, 20, 21, 22, 24, 25, 27, 28, 29, 33, 34, 35, 37, 38, 40, 41, 42, 45, 46, 47, 49, 50, 54, 55, 56, 58, 59, 61, 62, 63, 66, 67, 68, 70, 71, 74, 75, 76, 78, 79, 81, 82, 83, 88, 89, 90, 92, 93, 95, 96, 97, 100, 101, 102

Offset: 0

Antti Karttunen, Nov 24 2012

nonn

These are the positive integers i for which there exists k such that A007895(i+k)=k.

Starting offset is zero, because a(0) = 0 is a special case. Start indexing from 1 when you want only nonzero natural numbers satisfying the same condition.

Antti Karttunen, Table of n, a(n) for n = 0..10947

Cf. A007895, A022342, A219641. Complement: A219638. Union of A219639 and A219637.
First differences: A261095.
Characteristic function: A261092.
Left inverses: A261093, A261094.
Analogous sequences for other bases: A005187, A219650.

(define (A219640 n) (A219641 (A022342 (+ 1 n))))

a(n) = A219641(A022342(n+1)).
Other identities. For all n >= 0:
A261093(a(n)) = n.
A261094(a(n)) = n.

Starting offset changed to 0 by Antti Karttunen, Aug 08 2015

A219637 Numbers that occur twice in A219641.

Original entry on oeis.org

0, 2, 4, 7, 9, 12, 14, 16, 20, 22, 24, 27, 29, 33, 35, 37, 40, 42, 45, 47, 49, 54, 56, 58, 61, 63, 66, 68, 70, 74, 76, 78, 81, 83, 88, 90, 92, 95, 97, 100, 102, 104, 108, 110, 112, 115, 117, 121, 123, 125, 128, 130, 133, 135, 137, 143, 145, 147, 150, 152, 155

Offset: 1

Antti Karttunen, Nov 24 2012

nonn

A. Karttunen, Table of n, a(n) for n = 1..10000

Cf. A219636, A219639, A219641.

cp's OEIS Frontend

A219641 a(n) = n minus (number of 1's in Zeckendorf expansion of n).

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A219640 Numbers m for which there exists k such that m = k - (number of 1's in Zeckendorf expansion of k); distinct values in A219641.

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A219637 Numbers that occur twice in A219641.

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