A219645 -id:A219645 - cp's OEIS frontend

A261084 a(n) = A219645(n) - A219648(n).

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 0, 0, 1, 1, 2, 1, 2, 3, 0, 0, 1, 1, 2, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 2, 0, 0, 0, 1, 1, 0, 1, 2, 3, 3, 3, 4, 0, 0, 1, 1, 2, 0, 0, 0, 1, 1, 0, 1, 2, 3, 3, 3, 4, 2, 2, 1, 2, 3, 2, 3, 3, 0, 0, 1, 1, 2, 0, 0, 0, 1, 1, 0, 1, 2, 3, 3, 3, 4, 2, 2, 1, 2, 3, 2, 3, 3, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 2, 0, 0, 0

Offset: 0

Antti Karttunen, Aug 09 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..11817

Cf. A219648, A219645, A261083.

(define (A261084 n) (- (A219645 n) (A219648 n)))

a(n) = A219645(n) - A219648(n).

A219648 The infinite trunk of Zeckendorf beanstalk. The only infinite sequence such that a(n-1) = a(n) - number of 1's in Zeckendorf representation of a(n).

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 9, 12, 14, 17, 20, 22, 24, 27, 29, 33, 35, 37, 40, 42, 45, 47, 50, 54, 56, 58, 61, 63, 67, 70, 74, 76, 79, 83, 88, 90, 92, 95, 97, 101, 104, 108, 110, 113, 117, 121, 123, 126, 130, 134, 138, 143, 145, 147, 150, 152, 156, 159, 163, 165, 168

Offset: 0

Antti Karttunen, Nov 24 2012

nonn

a(n) tells in what number we end in n steps, when we start climbing up the infinite trunk of the "Zeckendorf beanstalk" from its root (zero).
There are many finite sequences such as 0,1,2; 0,1,2,4,5; etc. (see A219649) and as the length increases, so (necessarily) does the similarity to this infinite sequence.
There can be only one infinite trunk in "Zeckendorf beanstalk" as all paths downwards from numbers >= A000045(i) must pass through A000045(i)-1 (i.e. A000071(i)). This provides also a well-defined method to compute the sequence, for example, via a partially reversed version A261076.
See A014417 for the Fibonacci number system representation, also known as Zeckendorf expansion.

Antti Karttunen, Table of n, a(n) for n = 0..11817

Cf. A000045, A000071, A007895, A014417, A219641, A219649, A261076, A261102. For all n, A219642(a(n)) = n and A219643(n) <= a(n) <= A219645(n). Cf. also A261083 & A261084.

Other similarly constructed sequences: A179016, A219666, A255056.

(define (A219648 n) (A261076 (A261102 n)))

a(n) = A261076(A261102(n)).

A219642 Number of steps to reach 0 starting with n and using the iterated process: x -> x - (number of 1's in Zeckendorf expansion of x).

Original entry on oeis.org

0, 1, 2, 3, 3, 4, 4, 5, 6, 6, 7, 7, 7, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 21, 21, 21, 22, 22, 22, 22, 23, 24, 24, 25, 25, 25, 26, 26, 26, 27, 27, 27, 28, 28, 29, 29

Offset: 0

Antti Karttunen, Nov 24 2012

nonn

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

A. Karttunen, Table of n, a(n) for n = 0..10946

Cf. A007895, A014417, A219640, A219641, A219643-A219645, A219648. Analogous sequence for binary system: A071542, for factorial number system: A219652.

A007895(n)=if(n<4, n>0, my(k=2,s,t); while(fibonacci(k++)<=n,); while(k && n, t=fibonacci(k); if(t<=n, n-=t; s++); k--); s)
a(n)=my(s); while(n, n-=A007895(n); s++); s \\ Charles R Greathouse IV, Sep 02 2015

from sympy import fibonacci
def a007895(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")
def a219641(n): return n - a007895(n)
l=[0]
for n in range(1, 101):
    l.append(1 + l[a219641(n)])
print(l) # Indranil Ghosh, Jun 09 2017

a(0)=0; for n>0, a(n) = 1+a(A219641(n)).

A219655 Greatest inverse of A219652; a(n) = maximal i such that A219652(i) = n.

Original entry on oeis.org

0, 1, 3, 5, 7, 11, 15, 19, 23, 25, 29, 33, 37, 41, 47, 51, 55, 59, 65, 71, 77, 83, 89, 95, 101, 107, 115, 119, 121, 125, 129, 133, 137, 143, 147, 151, 155, 161, 167, 173, 179, 185, 191, 197, 203, 211, 217, 225, 233, 239, 243, 247, 251, 257, 263, 269, 275, 281

Offset: 0

Antti Karttunen, Nov 25 2012

nonn

A. Karttunen, Table of n, a(n) for n = 0..10080

Cf. A219653 for the least inverse. A219654 gives the first differences.

This sequence is based on factorial number system: A007623. Analogous sequence for binary system: A173601, for Zeckendorf expansion: A219645.

a(n) = A219653(n) + A219654(n) - 1.

A219643 Least inverse of A219642; a(n) = minimal i such that A219642(i) = n.

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 8, 10, 13, 15, 18, 21, 23, 26, 29, 32, 34, 36, 39, 42, 45, 47, 50, 54, 55, 57, 60, 63, 66, 68, 71, 75, 78, 81, 84, 89, 91, 94, 97, 100, 102, 105, 109, 112, 115, 118, 123, 126, 130, 134, 138, 143, 144, 146, 149, 152, 155, 157, 160, 164, 167

Offset: 0

Antti Karttunen, Nov 24 2012

nonn

A. Karttunen, Table of n, a(n) for n = 0..10000

Cf. A219645 for the greatest inverse. A219644 gives the first differences.

This sequence is based on Fibonacci number system (Zeckendorf expansion): A014417. For analogous sequence in binary system, see: A213708, for factorial number system: A219653.

A255055 Greatest inverse of A255072; a(n) = largest k such that A255072(k) = n.

Original entry on oeis.org

0, 2, 5, 6, 10, 13, 14, 18, 22, 26, 29, 30, 34, 38, 43, 46, 50, 54, 58, 61, 62, 66, 70, 75, 78, 85, 90, 94, 98, 102, 107, 110, 114, 118, 122, 125, 126, 130, 134, 139, 142, 149, 154, 158, 165, 171, 175, 181, 186, 190, 194, 198, 203, 206, 213, 218, 222, 226, 230, 235, 238, 242, 246, 250, 253, 254, 258

Offset: 0

Antti Karttunen, Feb 14 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16143

Cf. A255053, A255054, A255072.

Analogous sequences: A173601, A219645, A219655.

a(0) = 0; for n > 0, a(n) = A255054(n) + a(n-1).
Other identities. For all n >= 0:
a(n) = A255053(n) + A255054(n) - 1.
a(n) = A255056(n) + A255124(n).

A219644 Run lengths in A219642.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 3, 2, 3, 3, 2, 3, 3, 3, 2, 2, 3, 3, 3, 2, 3, 4, 1, 2, 3, 3, 3, 2, 3, 4, 3, 3, 3, 5, 2, 3, 3, 3, 2, 3, 4, 3, 3, 3, 5, 3, 4, 4, 4, 5, 1, 2, 3, 3, 3, 2, 3, 4, 3, 3, 3, 5, 3, 4, 4, 4, 5, 3, 3, 3, 5, 5, 3, 5, 5, 3, 2, 3, 3, 3, 2, 3, 4, 3, 3, 3

Offset: 0

Antti Karttunen, Nov 24 2012

nonn

a(n) tells from how many starting values one can end to 0 in n steps, with the iterative process described in A219642 (if going around in 0->0 loop is disallowed).

A. Karttunen, Table of n, a(n) for n = 0..10000

a(n) = 1+(A219645(n)-A219643(n)).

This sequence is based on Fibonacci number system (Zeckendorf expansion): A014417. Analogous sequence for binary system: A086876, for factorial number system: A219654.

a(n) = A219643(n+1)-A219643(n). (The first differences of A219643).

cp's OEIS Frontend

A261084 a(n) = A219645(n) - A219648(n).

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A219648 The infinite trunk of Zeckendorf beanstalk. The only infinite sequence such that a(n-1) = a(n) - number of 1's in Zeckendorf representation of a(n).

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A219642 Number of steps to reach 0 starting with n and using the iterated process: x -> x - (number of 1's in Zeckendorf expansion of x).

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A219655 Greatest inverse of A219652; a(n) = maximal i such that A219652(i) = n.

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A219643 Least inverse of A219642; a(n) = minimal i such that A219642(i) = n.

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A255055 Greatest inverse of A255072; a(n) = largest k such that A255072(k) = n.

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A219644 Run lengths in A219642.

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