A138703 -id:A138703 - 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

A138706 a(n) is the sum of the terms in the continued fraction expansion of the absolute value of B_{2n}, the (2n)-th Bernoulli number.

Original entry on oeis.org

1, 6, 30, 42, 30, 18, 37, 7, 28, 96, 559, 6210, 86617, 1425523, 27298263, 601580913, 15116315788, 429614643067, 13711655205344, 488332318973599, 19296579341940107, 841693047573684421, 40338071854059455479, 2115074863808199160579, 120866265222965259346062

Offset: 0

Leroy Quet, Mar 26 2008

nonn

			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 a(6) = 0 + 3 + 1 + 19 + 3 + 11 = 37.

Cf. A027641/A027642, A138703, A138704, A138705.

A138704row := proc(n) local B; B := abs(bernoulli(2*n)) ; numtheory[cfrac](B,20,'quotients') ; end: A138706 := proc(n) add(c,c=A138704row(n)) ; end: seq(op(A138706(n)),n=0..30) ; # R. J. Mathar, Jul 20 2009

Table[Total[ContinuedFraction[Abs[BernoulliB[2n]]]],{n,0,25}] (* Harvey P. Dale, Feb 23 2012 *)

a(n) = vecsum(contfrac(abs(bernfrac(2*n)))); \\ Jinyuan Wang, Aug 07 2021

from sympy import continued_fraction, bernoulli
def A138706(n): return sum(continued_fraction(abs(bernoulli(n<<1)))) # Chai Wah Wu, Apr 14 2023

a(n) = A138703(2*n). - R. J. Mathar, Jul 20 2009

a(7)-a(22) from R. J. Mathar, Jul 20 2009

More terms from Jinyuan Wang, Aug 07 2021

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

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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A138706 a(n) is the sum of the terms in the continued fraction expansion of the absolute value of B_{2n}, the (2n)-th Bernoulli number.

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

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PARI

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