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A157168
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Denominators of partial sums of a series related to Lebesgue's constant L(1) = (1 + 6*sqrt(3)/Pi)/3 approximately 1.435991124.
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3
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1, 1155, 1786785, 2342475135, 448349740839, 903424727790585, 6852476560291587225, 47086107436387612457865, 1354734576812637878430573, 19582688307826680532713932715, 3924930242268690398199663942735, 66681761294543429372270719198537125, 102223140064535077227691012531357412625
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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).
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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.
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EXAMPLE
| Rationals R(1;n): [23, 33073/1155, 55943738/1786785, 77064019958/2342475135,...].
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CROSSREFS
| A157165/A157166 related to L(0) = 1.
Sequence in context: A088012 A046406 A136355 * A062915 A034979 A189279
Adjacent sequences: A157165 A157166 A157167 * A157169 A157170 A157171
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KEYWORD
| nonn,frac,easy
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de) Oct 16 2009
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