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Search: id:A157167
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| A157167 |
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Numerators 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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+0
4
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| 23, 33073, 55943738, 77064019958, 15226093370063, 31370562345762421, 241905492960111168964, 1683591136668277300660676, 48935652383592600478507247, 713289082617826259771761324613, 143961819529547244077111055694498, 2460282354560331257420364974778935366
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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For the denominators see A157168.
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) a(n)/A157168(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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REFERENCES
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G. Szego, \"Uber die Lebesgueschen Konstanten bei den Fourierschen Reihen, Math. Z. 9 (1921) 163-166.
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LINKS
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W. Lang: Sequences related to Lebesgue's constants.
Mathworld: Lebesgue Constants.
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FORMULA
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a(n) = numerator(R(1;n)) = numerator(45*sum(Theta(1,3*k)/(4*k^2-1),k=1..n)), n>=1.
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EXAMPLE
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Rationals R(1;n): [23, 33073/1155, 55943738/1786785, 77064019958/2342475135,...].
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CROSSREFS
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A157165/A157166 related to L(0) = 1.
Adjacent sequences: A157164 A157165 A157166 this_sequence A157168 A157169 A157170
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KEYWORD
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nonn,frac,easy,new
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang@physik.uni-karlsruhe.de) Oct 16 2009, Nov 24 2009
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