A157168 Denominators of partial sums of a series related to Lebesgue's constant L(1) = (1 + 6*sqrt(3)/Pi)/3 approximately 1.435991124.

1, 1155, 1786785, 2342475135, 448349740839, 903424727790585, 6852476560291587225, 47086107436387612457865, 1354734576812637878430573, 19582688307826680532713932715, 3924930242268690398199663942735, 66681761294543429372270719198537125, 102223140064535077227691012531357412625

Offset

1,2

Comments

For the numerators see A157167.

Lebesgue's constants L(n):= (2/Pi)*int(|sin((2*n+1)*x)|/sin(x),x=0..Pi/2). (Called \rho_n in the Szego reference). L(1) = (1 + 6*sqrt(3)/Pi)/3.

L(1) = (16/(Pi^2))*sum(Theta(1,3*k)/(4*k^2-1),k=1..infty) with Theta(1,m):=sum(1/(2*j-1),j=1..m) = int(((sin(m*x))^2)/sin(x),x=0..Pi/2) (see Szego reference formula (R), p.165 and the line before this).

The rationals (partial sums) R(1;n):=45*sum(Theta(1,3*k)/(4*k^2-1),k=1..n) give (in lowest terms) A157167(n)/a(n). The sequence {R(1;n)/45} converges slowly to ((Pi^2)/48)*(1 + 6*sqrt(3)/Pi), approximately 0.8857915201 because of the given L(1) value (see the W. Lang link for R(1;10^n)/45 for n=0..4).

Links

Table of n, a(n) for n=1..13.

Formula

a(n) = denominator(R(1;n)) = denominator(45*sum(Theta(1,3*k)/(4*k^2-1),k=1..n)), n>=1, with Theta(1,m) defined above.

Example

Rationals R(1;n): [23, 33073/1155, 55943738/1786785, 77064019958/2342475135,...].

Mathematica

theta[1, k_] := Sum[1/(2*j-1), {j, 1, k}]; a[n_] := Denominator[45*Sum[theta[1, 3*k]/(4*k^2-1), {k, 1, n}]]; Table[a[n], {n, 1, 13}] (* Jean-François Alcover, Dec 02 2013 *)

Crossrefs

A157165/A157166 related to L(0) = 1.

Sequence in context: A284846 A136355 A321245 * A062915 A225938 A221813

Adjacent sequences: A157165 A157166 A157167 * A157169 A157170 A157171

Keyword

nonn,frac,easy

Author

Wolfdieter Lang, Oct 16 2009

Status

approved