A138701 -id:A138701 - cp's OEIS frontend

A138703 a(n) is the sum of the terms in the continued fraction of the absolute value of B_n, the n-th Bernoulli number.

1, 2, 6, 0, 30, 0, 42, 0, 30, 0, 18, 0, 37, 0, 7, 0, 28, 0, 96, 0, 559, 0, 6210, 0, 86617, 0, 1425523, 0, 27298263, 0, 601580913, 0, 15116315788, 0, 429614643067, 0, 13711655205344, 0, 488332318973599, 0, 19296579341940107, 0, 841693047573684421, 0, 40338071854059455479

Offset: 0

Leroy Quet, Mar 26 2008

nonn

For all odd n >=3, a(n) = 0.

			The 12th Bernoulli number is -691/2730. Now 691/2730 = the continued fraction 0 + 1/(3 + 1/(1 + 1/(19 + 1/(3 + 1/11)))). So a(12) = 0 + 3 + 1 + 19 + 3 + 11 = 37.

Jinyuan Wang, Table of n, a(n) for n = 0..635

Cf. A027641/A027642, A138701, A138702, A138706.

A138701row := proc(n) local B; B := abs(bernoulli(n)) ; numtheory[cfrac](B,20,'quotients') ; end: A138703 := proc(n) add(c,c=A138701row(n)) ; end: seq(op(A138703(n)),n=0..80) ; # R. J. Mathar, Jul 20 2009

Table[ ContinuedFraction[ BernoulliB[n] // Abs] // Total, {n, 0, 50}] (* Jean-François Alcover, Mar 27 2013 *)

a(n) = vecsum(contfrac(abs(bernfrac(n)))); \\ Jinyuan Wang, Aug 07 2021

from sympy import continued_fraction, bernoulli
def A138703(n): return sum(continued_fraction(abs(bernoulli(n)))) # Chai Wah Wu, Apr 14 2023

Extended beyond a(15) by R. J. Mathar, Jul 20 2009

More terms from Jean-François Alcover, Mar 27 2013

A138704 Irregular array read by rows: row n contains the continued fraction terms (in order) for the absolute value of B_{2n}, the (2n)th Bernoulli number.

Original entry on oeis.org

1, 0, 6, 0, 30, 0, 42, 0, 30, 0, 13, 5, 0, 3, 1, 19, 3, 11, 1, 6, 7, 10, 1, 5, 1, 2, 2, 54, 1, 33, 1, 2, 3, 2, 529, 8, 20, 2, 6192, 8, 8, 2, 86580, 3, 1, 19, 3, 11, 1425517, 6, 27298231, 14, 1, 2, 1, 14, 601580873, 1, 9, 15, 2, 7, 6, 15116315767, 10, 1, 5, 1, 2, 2, 429614643061, 6

Offset: 0

Leroy Quet, Mar 26 2008

The number of terms in row n is A138705(n).

			The 12th Bernoulli number is -691/2730. Now 691/2730 has the continued fraction 0 + 1/(3 + 1/(1 + 1/(19 + 1/(3 + 1/11)))). So row 6 is (0,3,1,19,3,11).

Michael De Vlieger, Table of n, a(n) for n = 0..2884

Cf. A138701, A138705, A138706, A000367, A002445.

A138704row := proc(n) local B; B := abs(bernoulli(2*n)) ; numtheory[cfrac](B,20,'quotients') ; end: seq(op(A138704row(n)),n=0..20) ; # R. J. Mathar, Jul 20 2009

Array[ContinuedFraction@ Abs@ BernoulliB[2 #] &, 18, 0] // Flatten (* Michael De Vlieger, Oct 18 2017 *)

More terms from R. J. Mathar, Jul 20 2009

A138702 a(n) = number of terms in the continued fraction of the absolute value of B_n, the n-th Bernoulli number.

Original entry on oeis.org

1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 6, 1, 2, 1, 7, 1, 7, 1, 4, 1, 4, 1, 6, 1, 2, 1, 6, 1, 7, 1, 7, 1, 2, 1, 10, 1, 2, 1, 8, 1, 2, 1, 3, 1, 5, 1, 10, 1, 3, 1, 7, 1, 7, 1, 6, 1, 6, 1, 17, 1, 2, 1, 7, 1, 10, 1, 2, 1, 7, 1, 23, 1, 2, 1, 2, 1, 5, 1, 18, 1, 5, 1, 16, 1, 2, 1, 10, 1, 14, 1, 6, 1, 2, 1, 18, 1, 2, 1

Offset: 0

Leroy Quet, Mar 26 2008

nonn

The continued fraction terms being counted include the initial 0, if there is one. (a(n), for all odd n >= 3, is 1.)

			The 12th Bernoulli number is -691/2730. Now 691/2730 has the continued fraction 0 + 1/(3 + 1/(1 + 1/(19 + 1/(3 + 1/11)))), which has 6 terms (including the zero). So a(12) = 6.

Antti Karttunen, Table of n, a(n) for n = 0..16383

Cf. A138701, A138703.

lcf(x)=local(r);r=1;while(1,x-=floor(x);if(x==0,return(r));x=1/x;r++)
a(n)=lcf(abs(bernfrac(n))) \\ Franklin T. Adams-Watters, May 14 2010

More terms from Franklin T. Adams-Watters, May 14 2010

cp's OEIS Frontend

A138703 a(n) is the sum of the terms in the continued fraction of the absolute value of B_n, the n-th Bernoulli number.

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A138704 Irregular array read by rows: row n contains the continued fraction terms (in order) for the absolute value of B_{2n}, the (2n)th Bernoulli number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A138702 a(n) = number of terms in the continued fraction of the absolute value of B_n, the n-th Bernoulli number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions