A109170 -id:A109170 - cp's OEIS frontend

A109169 Decimal expansion of constant x such that the continued fraction expansion of 2*x (A109170) yields the continued fraction expansion of x (A109168) interleaved with positive even numbers.

1, 4, 0, 8, 4, 9, 4, 2, 7, 9, 2, 2, 8, 9, 0, 6, 9, 8, 5, 7, 4, 8, 4, 7, 4, 2, 7, 9, 0, 8, 0, 6, 9, 7, 9, 9, 1, 6, 1, 3, 9, 9, 8, 9, 5, 5, 7, 8, 2, 0, 5, 1, 2, 8, 1, 4, 6, 6, 2, 6, 3, 8, 1, 7, 5, 2, 4, 8, 6, 2, 9, 7, 7, 8, 9, 9, 0, 3, 0, 8, 5, 3, 3, 0, 1, 2, 5, 6, 2, 8, 5, 4, 3, 0, 4, 8, 6, 9, 1, 8, 6, 4, 8, 1, 2

Offset: 1

Paul D. Hanna, Jun 21 2005

			x=1.408494279228906985748474279080697991613998955782051281466263817524862977...
The continued fraction expansion of x = A109168:
[1; 2, 2, 4, 3, 4, 4, 8, 5, 6, 6, 8, 7, 8, 8, 16, ...];
the continued fraction expansion of 2*x = A109170:
[2;1, 4,2, 6,2, 8,4, 10,3, 12,4, 14,4, 16,8, 18,5, ...]
which equals the continued fraction of x interleaved with even numbers.

Cf. A109168 (continued fraction of x), A109170 (continued fraction of 2*x), A109171 (digits of 2*x).

{PQ(n)=if(n%2==1,(n+1)/2,2*PQ(n/2))}
{CFM=contfracpnqn(vector(500,n,PQ(n))); CFM[1,1]/CFM[2,1]*1.0}

A109171 Decimal expansion of 2x, where constant x (A109169) satisfies the condition that the continued fraction expansion of 2x (A109170) is equal to the continued fraction expansion of x (A109168) interleaved with positive even numbers.

Original entry on oeis.org

2, 8, 1, 6, 9, 8, 8, 5, 5, 8, 4, 5, 7, 8, 1, 3, 9, 7, 1, 4, 9, 6, 9, 4, 8, 5, 5, 8, 1, 6, 1, 3, 9, 5, 9, 8, 3, 2, 2, 7, 9, 9, 7, 9, 1, 1, 5, 6, 4, 1, 0, 2, 5, 6, 2, 9, 3, 2, 5, 2, 7, 6, 3, 5, 0, 4, 9, 7, 2, 5, 9, 5, 5, 7, 9, 8, 0, 6, 1, 7, 0, 6, 6, 0, 2, 5, 1, 2, 5, 7, 0, 8, 6, 0, 9, 7, 3, 8, 3, 7, 2, 9, 6, 2, 5

Offset: 1

Paul D. Hanna, Jun 21 2005

			2*x=2.8169885584578139714969485581613959832279979115641025629325276350497259...
The continued fraction expansion of x = A109168:
[1; 2, 2, 4, 3, 4, 4, 8, 5, 6, 6, 8, 7, 8, 8, 16, ...];
the continued fraction expansion of 2*x = A109170:
[2;1, 4,2, 6,2, 8,4, 10,3, 12,4, 14,4, 16,8, 18,5, ...]
which equals the continued fraction of x interleaved with even numbers.

Cf. A109168 (continued fraction of x), A109169 (digits of x), A109170 (continued fraction of 2*x).

{PQ(n)=if(n%2==1,(n+1)/2,2*PQ(n/2))}
{CFM=contfracpnqn(vector(500,n,PQ(n))); x2=CFM[1,1]/CFM[2,1]*2.0}

A109168 Continued fraction expansion of the constant x (A109169) such that the continued fraction of 2*x yields the continued fraction of x interleaved with the positive even numbers.

Original entry on oeis.org

1, 2, 2, 4, 3, 4, 4, 8, 5, 6, 6, 8, 7, 8, 8, 16, 9, 10, 10, 12, 11, 12, 12, 16, 13, 14, 14, 16, 15, 16, 16, 32, 17, 18, 18, 20, 19, 20, 20, 24, 21, 22, 22, 24, 23, 24, 24, 32, 25, 26, 26, 28, 27, 28, 28, 32, 29, 30, 30, 32, 31, 32, 32, 64, 33, 34, 34, 36, 35, 36, 36, 40, 37, 38, 38

Offset: 1

Paul D. Hanna, Jun 21 2005

Compare with continued fraction A100338.

The sequence is equal to the sequence of positive integers (1, 2, 3, 4, ...) interleaved with the sequence multiplied by two, 2*(1, 2, 2, 4, 3, ...) = (2, 4, 4, 8, 6, ...): see the first formula. - M. F. Hasler, Oct 19 2019

			x=1.408494279228906985748474279080697991613998955782051281466263817524862977...
The continued fraction expansion of 2*x = A109170:
[2;1, 4,2, 6,2, 8,4, 10,3, 12,4, 14,4, 16,8, 18,5, ...]
which equals the continued fraction of x interleaved with the even numbers.

Cf. A109169 (digits of x), A109170 (continued fraction of 2*x), A109171 (digits of 2*x).
Cf. A006519 and A129760. [Johannes W. Meijer, Jun 22 2011]
Half the terms of A285326.

nmax:=75; pmax:= ceil(log(nmax)/log(2)); for p from 0 to pmax do for n from 1 to nmax do a((2*n-1)*2^p):= n*2^p: od: od: seq(a(n), n=1..nmax); # Johannes W. Meijer, Jun 22 2011

PARI
```
a(n)=if(n%2==1,(n+1)/2,2*a(n/2))
```

A109168(n)=(n+bitand(n,-n))\2 \\ M. F. Hasler, Oct 19 2019

;; With memoization-macro definec
(definec (A109168 n) (if (zero? n) n (if (odd? n) (/ (+ 1 n) 2) (* 2 (A109168 (/ n 2))))))
;; Antti Karttunen, Apr 19 2017

a(2*n-1) = n, a(2*n) = 2*a(n) for all n >= 1.
a((2*n-1)*2^p) = n * 2^p, p >= 0. - Johannes W. Meijer, Jun 22 2011
a(n) = n - (n AND n-1)/2. - Gary Detlefs, Jul 10 2014
a(n) = A285326(n)/2. - Antti Karttunen, Apr 19 2017
a(n) = A140472(n). - M. F. Hasler, Oct 19 2019

cp's OEIS Frontend

A109169 Decimal expansion of constant x such that the continued fraction expansion of 2*x (A109170) yields the continued fraction expansion of x (A109168) interleaved with positive even numbers.

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A109171 Decimal expansion of 2x, where constant x (A109169) satisfies the condition that the continued fraction expansion of 2x (A109170) is equal to the continued fraction expansion of x (A109168) interleaved with positive even numbers.

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A109168 Continued fraction expansion of the constant x (A109169) such that the continued fraction of 2*x yields the continued fraction of x interleaved with the positive even numbers.

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cp's OEIS Frontend

A109169 Decimal expansion of constant x such that the continued fraction expansion of 2*x (A109170) yields the continued fraction expansion of x (A109168) interleaved with positive even numbers.

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Author

Keywords

Examples

Crossrefs

Programs

PARI

A109171 Decimal expansion of 2*x, where constant x (A109169) satisfies the condition that the continued fraction expansion of 2*x (A109170) is equal to the continued fraction expansion of x (A109168) interleaved with positive even numbers.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

A109168 Continued fraction expansion of the constant x (A109169) such that the continued fraction of 2*x yields the continued fraction of x interleaved with the positive even numbers.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

PARI

PARI

Scheme

Formula

A109171 Decimal expansion of 2x, where constant x (A109169) satisfies the condition that the continued fraction expansion of 2x (A109170) is equal to the continued fraction expansion of x (A109168) interleaved with positive even numbers.