A219652 -id:A219652 - cp's OEIS frontend

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).

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)).

A255072 Number of steps to reach 0 starting with n and using the iterated process: x -> x - (number of runs in binary representation of x).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Feb 14 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8192

Cf. A255053 (least inverse), A255055 (greatest inverse), A255054 (run lengths).
Cf. A005811, A236840, A255071, A255056.
Cf. A255061 & A255062 (values at points (2^n)-2 and (2^n)-1).
Analogous sequences: A071542, A219642, A219652

a(0) = 0; for n >= 1, a(n) = 1 + a(A236840(n)) = 1 + a(n - A005811(n)).
Other identities. For all n >= 0:
a(A255053(n)) = a(A255055(n)) = n.
a(A255056(n)) = n. [This sequence works also as an inverse function for number-of-runs beanstalk A255056.]

A219665 One more than the partial sums of A219661.

Original entry on oeis.org

1, 2, 4, 9, 28, 111, 539, 3150, 21623, 172349, 1549897, 15401145, 168011253, 2003304294, 25928878273, 361788001016, 5411160126368, 86353882249912, 1464841397585336, 26323224850512720, 499551889319197566

Offset: 1

Antti Karttunen, May 28 2013

nonn

Are there any cases after n>2, for which A219666(a(n)) = n! instead of n!+1 ? (At least for all terms a(3) - a(14) that number is n!+1.)

Compare to the conjecture given at A213710.

One more than A226061.
Cf. A219652, A219666, A226061.
Cf. also A213710 (analogous sequence for base-2).

Accumulate@ Table[Length@ NestWhileList[# - Total@ IntegerDigits[#, MixedRadix[Reverse@ Range[2, 120]]] &, (n + 1)! - 1, # > n! - 1 &] - 1, {n, 0, 8}] + 1 (* Michael De Vlieger, Jun 27 2016, Version 10.2 *)

Scheme
```
(define (A219665 n) (+ 1 (A226061 n)))
```

a(n) = A226061(n)+1 = A219652(n!).

Terms a(16) - a(21) computed from the new terms of A219661 by Antti Karttunen, Jun 27 2016

A230420 Triangle T(n,k) giving the number of terms of A219666 which have n digits (A084558) in their factorial base expansion and whose most significant digit (A099563) in that base is k.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 6, 5, 4, 4, 22, 19, 16, 14, 12, 94, 82, 73, 65, 59, 55, 479, 432, 395, 362, 336, 314, 293, 2886, 2667, 2482, 2324, 2189, 2073, 1971, 1881, 20276, 19123, 18124, 17249, 16473, 15775, 15140, 14555, 14011, 164224, 156961, 150389, 144378, 138828, 133664, 128831, 124289, 120010, 115974

Offset: 1

Antti Karttunen, Oct 18 2013

See A007623 for the factorial number system representation.

			The first rows of this triangular table are:
1;
1, 1;
2, 2, 1;
6, 5, 4, 4;
22, 19, 16, 14, 12;
94, 82, 73, 65, 59, 55;
...
T(4,2) = 5 as only the terms 48, 52, 57, 63 and 70 of A219666 (with factorial base representations 2000, 2020, 2111, 2211 and 2320) have four significant digits in the factorial base, with the most significant digit being 2.

Transpose: A230421. Row sums: A219661. Cf. also A230428, A230429, A219652, A219666.

(define (A230420 n) (if (<= n 3) 1 (let loop ((i (A230429 n)) (s 0)) (cond ((not (= (A099563 i) (A002260 n))) s) (else (loop (A219651 i) (+ 1 s)))))))

T(n,k) = 1 + A219652(A230429(n,k)) - A219652(A230428(n,k)).

cp's OEIS Frontend

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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A255072 Number of steps to reach 0 starting with n and using the iterated process: x -> x - (number of runs in binary representation of x).

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A219665 One more than the partial sums of A219661.

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A230420 Triangle T(n,k) giving the number of terms of A219666 which have n digits (A084558) in their factorial base expansion and whose most significant digit (A099563) in that base is k.

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