A317332 -id:A317332 - cp's OEIS frontend

A317335 a(n) = A317332(n) - 8*n.

1, 1, -2, 1, 1, -2, -2, 1, 1, 1, -2, -2, 1, -2, -2, 1, 1, 1, -2, 1, 1, -2, -2, -2, 1, 1, -2, -2, 1, -2, -2, 1, 1, 1, -2, 1, 1, -2, -2, 1, 1, 1, -2, -2, 1, -2, -2, -2, 1, 1, -2, 1, 1, -2, -2, -2, 1, 1, -2, -2, 1, -2, -2

Offset: 1

A.H.M. Smeets, Jul 26 2018

sign

A.H.M. Smeets, Table of n, a(n) for n = 1..20000

Cf. A317331, A317332, A317333, A317336.

n, f, i, p, q = 1, 1, 0, 0, 1
while i < 1000000:
    i, p, q = i+1, p*10, q*10
    if i == f:
        p, n = p+1, n+1
        f = f*n
n, a, j = 0, 0, 0
while p%q > 0:
    a, f, p, q = a+1, p//q, q, p%q
    if f == 8:
        n = n+1
        print(n, a-1-8*n)

a(4*n-3) = 1, a(4*n-1) = -2, a(2*n) = a(n) for n > 0.
abs(a(n)+1) = A014710(n-1).
abs(a(n)) = A014709(n-1).

A058304 Continued fraction for Liouville's number (A012245).

Original entry on oeis.org

0, 9, 11, 99, 1, 10, 9, 999999999999, 1, 8, 10, 1, 99, 11, 9, 999999999999999999999999999999999999999999999999999999999999999999999999, 1, 8, 11, 99, 1, 10, 8, 1, 999999999999, 9, 10, 1, 99, 11, 9

Offset: 0

Robert G. Wilson v, Dec 08 2000

From A.H.M. Smeets, Jun 06 2018: (Start)
Except for the first term, the only values that occur in this sequence are 1,8,9,10,11,and values 10^((m-1)*m!)-1 for m > 1. The probability of occurrence P(a(n) = k) are given by:
P(a(n) = 1) = 1/4,
P(a(n) = 8) = 1/8,
P(a(n) = 9) = 1/8,
P(a(n) = 10) = 1/8,
P(a(n) = 11) = 1/8 and
P(a(n) = 10^((m-1)*m!)-1) = 2^-(m+1) for m > 1. (End)

			0.1100010000000000000000010... = 0 + 1/(9 + 1/(11 + 1/(99 + 1/(1 + ...)))). - _Harry J. Smith_, May 15 2009

Harold M. Stark, "An Introduction to Number Theory," The MIT Press, Cambridge, MA and London, England, Eighth Printing, 1994, pages 172 - 177.

Muniru A Asiru, Table of n, a(n) for n = 0..62
J. O. Shallit, Simple Continued Fractions for Some Irrational Numbers II, J. Number Theory 14 (1982), 228-231.
Eric Weisstein's World of Mathematics, Liouville's Constant
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A012245.

Cf. A317413 (in base 2), A317414 (in base 3) A317661 (in base 4 and general).

with(numtheory): cfrac(add(1/10^factorial(n),n=1..7),62,'quotients'); # Muniru A Asiru, Aug 08 2018

ContinuedFraction[ Sum[ 1 /10^(n!), {n, 1, 7} ], 40 ]

{ allocatemem(932245000); default(realprecision, 200000); x=contfrac(suminf(n=1, 1.0/10^n!)); for (n=1, 255, write("b058304.txt", n, " ", x[n])); } \\ Harry J. Smith, May 15 2009

n,f,i,p,q,base = 1,1,0,0,1,10
while i < 1000:
    i,p,q = i+1,p*base,q*base
    if i == f:
        p,n = p+1,n+1
        f = f*n
n,a,j = 0,0,0
while p%q > 0:
    a,f,p,q = a+1,p//q,q,p%q
    print(a-1,f)
# A.H.M. Smeets, Aug 03 2018

From A.H.M. Smeets, Jun 26 2018: (Start)
a(n) = 1 iff n in A317331,
a(n) = 8 iff n in A317332,
a(n) = 9 iff n in A317333,
a(n) = 10 iff n = 8*m - 6 + 3*(m mod 2) for m > 0,
a(n) = 11 iff n = 8*m - 3 - 3*(m mod 2) for m > 0,
a(n) = 10^((m-1)*m!)-1 iff n in {2^m*(1+k*4) - 1 | k >= 0} union {2^m*(3+k*4) | k >= 0} for m > 1. (End)

Offset changed to 0 on the advice of A.H.M. Smeets by Muniru A Asiru, Aug 11 2018

A317331 Indices m for which A058304(m) = 1.

Original entry on oeis.org

4, 8, 11, 16, 20, 23, 27, 32, 36, 40, 43, 47, 52, 55, 59, 64, 68, 72, 75, 80, 84, 87, 91, 95, 100, 104, 107, 111, 116, 119, 123, 128, 132, 136, 139, 144, 148, 151, 155, 160, 164, 168, 171, 175, 180, 183, 187, 191, 196, 200, 203, 208, 212, 215, 219, 223, 228, 232, 235, 239, 244, 247, 251

Offset: 1

A.H.M. Smeets, Jul 26 2018

nonn

A.H.M. Smeets, Table of n, a(n) for n = 1..20000

Cf. A014577, A058304, A317332, A317333, A317335, A317336.

n, f, i, p, q = 1, 1, 0, 0, 1
while i < 100000:
    i, p, q = i+1, p*10, q*10
    if i == f:
        p, n = p+1, n+1
        f = f*n
n, a, j = 0, 0, 0
while p%q > 0:
    a, f, p, q = a+1, p//q, q, p%q
    if f == 1:
        n = n+1
        print(n, a-1)

a(n) = 4*n + A014577(n-1)-1. - A.H.M. Smeets, Jul 29 2018

A317333 Indices m for which A058304(m) = 9.

Original entry on oeis.org

1, 6, 14, 25, 30, 38, 49, 57, 62, 70, 78, 89, 97, 102, 113, 121, 126, 134, 142, 153, 158, 166, 177, 185, 193, 198, 206, 217, 225, 230, 241, 249, 254, 262, 270, 281, 286, 294, 305, 313, 318, 326, 334, 345, 353, 358, 369, 377, 385, 390, 398, 409, 414, 422, 433, 441, 449, 454, 462, 473, 481, 486, 497, 505

Offset: 1

A.H.M. Smeets, Jul 26 2018

nonn

A.H.M. Smeets, Table of n, a(n) for n = 1..20000

Cf. A058304, A317331, A317332, A317335, A317336.

n, f, i, p, q = 1, 1, 0, 0, 1
while i < 1000000:
    i, p, q = i+1, p*10, q*10
    if i == f:
        p, n = p+1, n+1
        f = f*n
n, a, j = 0, 0, 0
while p%q > 0:
    a, f, p, q = a+1, p//q, q, p%q
    if f == 9:
        n = n+1
        print(n, a-1)

a(n) = 8*n + A317336(n).

A317336 a(n) = A317333(n) - 8*n.

Original entry on oeis.org

-7, -10, -10, -7, -10, -10, -7, -7, -10, -10, -10, -7, -7, -10, -7, -7, -10, -10, -10, -7, -10, -10, -7, -7, -7, -10, -10, -7, -7, -10, -7, -7, -10, -10, -10, -7, -10, -10, -7, -7, -10, -10, -10, -7, -7, -10, -7, -7, -7, -10, -10, -7, -10, -10, -7, -7, -7, -10, -10, -7, -7, -10, -7, -7

Offset: 1

A.H.M. Smeets, Jul 26 2018

sign

A.H.M. Smeets, Table of n, a(n) for n = 1..20000
Index entries for sequences obtained by enumerating foldings

Cf. A014710, A082410, A317331, A317332, A317333, A317335.

n, f, i, p, q = 1, 1, 0, 0, 1
while i < 1000000:
    i, p, q = i + 1, p * 10, q * 10
    if i == f:
        p, n = p + 1, n + 1
        f = f * n
n, a, j = 0, 0, 0
while p % q > 0:
    a, f, p, q = a + 1, p // q, q, p % q
    if f == 9:
        n = n + 1
        print(n, a - 1 - 8 * n)

a(4*n+4) = -7, a(4*n+2) = -10, for n > 0. a(1) = -7 and a(2*n-1) = a(n) for n > 1.
abs(a(n+2)+8) = A014710(n) for n >= 0.
a(n) = -7-3*A082410(n)

A317661 Continued fraction for quaternary expansion of Liouville's number interpreted in base 4 (A012245).

Original entry on oeis.org

0, 3, 5, 15, 1, 4, 3, 16777215, 1, 2, 4, 1, 15, 5, 3, 22300745198530623141535718272648361505980415, 1, 2, 5, 15, 1, 4, 2, 1, 16777215, 3, 4, 1, 15, 5, 3

Offset: 0

A.H.M. Smeets, Aug 03 2018

A.H.M. Smeets, Table of n, a(n) for n = 0..62

Cf. A012245, A317331, A317332, A317333.

Cf. A058304 (in base 10), A317413 (in base 2), A317414 (in base 3).

with(numtheory): cfrac(add(1/4^factorial(n),n=1..7),30,'quotients'); # Muniru A Asiru, Aug 12 2018

n, f, i, p, q, base = 1, 1, 0, 0, 1, 4
while i < 100000:
    i, p, q = i+1, p*base, q*base
    if i == f:
        p, n = p+1, n+1
        f = f*n
n, a, j = 0, 0, 0
while p%q > 0:
    a, f, p, q = a+1, p//q, q, p%q
    print(a-1, f)

In general for any Liouville's number base > 2:
a(n) = 1 if (and only if, for base > 3) n in A317331,
a(n) = base-2 if (and only if, for base > 3) n in A317332,
a(n) = base-1 if and only if n in A317333,
a(n) = base if and only if n in {8*m - 6 + 3*(m mod 2) | m > 0},
a(n) = base+1 if and only if n in {8*m - 3 - 3*(m mod 2) | m > 0},
a(n) = base^((m-1)*m!)-1 iff n in {2^m*(1+k*4) - 1 | k >= 0} union {2^m*(3+k*4) | k >= 0} for m > 1.

A359457 Continued fraction for constant A359456.

Original entry on oeis.org

0, 9, 11, 99, 1, 10, 9, 999999999999999999, 1, 8, 10, 1, 99, 11, 9

Offset: 0

A.H.M. Smeets, Jan 02 2023

It seems that all terms smaller than 100 are identical to the continued fraction terms of Liouville's constant as in A058304.
Except for the first term, the only values that occur in this sequence are 1,8,9,10,11,and values 10^A359458(m-1)-1 for m > 1. The probability of occurrence P(a(n) = k) are given by:
P(a(n) = 1) = 1/4,
P(a(n) = 8) = 1/8,
P(a(n) = 9) = 1/8,
P(a(n) = 10) = 1/8,
P(a(n) = 11) = 1/8 and
P(a(n) = 10^A359458(m-1)-1) = 2^-(m+1) for m > 1.

Cf. A058304, A317331, A317332, A317333, A359456, A359458.

a(n) = 1 if and only if n in A317331,
a(n) = 8 if and only if n in A317332,
a(n) = 9 if and only if n in A317333,
a(n) = 10 if and only if n = 8*m - 6 + 3*(m mod 2) for m > 0,
a(n) = 11 if and only if n = 8*m - 3 - 3*(m mod 2) for m > 0,
a(n) = 10^A359458(m-1)-1 if and only if n in {2^m*(1+k*4) - 1 | k >= 0} union {2^m*(3+k*4) | k >= 0} for m > 1.

cp's OEIS Frontend

A317335 a(n) = A317332(n) - 8*n.

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A058304 Continued fraction for Liouville's number (A012245).

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A317331 Indices m for which A058304(m) = 1.

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A317333 Indices m for which A058304(m) = 9.

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A317336 a(n) = A317333(n) - 8*n.

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A317661 Continued fraction for quaternary expansion of Liouville's number interpreted in base 4 (A012245).

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A359457 Continued fraction for constant A359456.

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