A138702 -id:A138702 - cp's OEIS frontend

A138701 Irregular array read by rows: row n contains the continued fraction terms (in order) for the absolute value of B_n, the n-th Bernoulli number.

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

Offset: 0

Leroy Quet, Mar 26 2008

Row n, for all odd n >= 3, is (0).

The number of terms in row n is A138702(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 12 is (0,3,1,19,3,11).

Michael De Vlieger, Table of n, a(n) for n = 0..3203

Cf. A138702, A138703, A138704, A027641, A027642.

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

Array[ContinuedFraction@ Abs@ BernoulliB@ # &, 31, 0] // Flatten (* Michael De Vlieger, Oct 18 2017 *)

Extended by R. J. Mathar, Jul 20 2009

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.

Original entry on oeis.org

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

cp's OEIS Frontend

A138701 Irregular array read by rows: row n contains the continued fraction terms (in order) for the absolute value of B_n, the n-th Bernoulli number.

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