A157167 Numerators of partial sums of a series related to Lebesgue's constant L(1) = (1 + 6*sqrt(3)/Pi)/3, approximately 1.435991124.

23, 33073, 55943738, 77064019958, 15226093370063, 31370562345762421, 241905492960111168964, 1683591136668277300660676, 48935652383592600478507247, 713289082617826259771761324613, 143961819529547244077111055694498, 2460282354560331257420364974778935366

Offset

1,1

Comments

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).

Links

Table of n, a(n) for n=1..12.

Wolfdieter Lang, Sequences related to Lebesgue's constants

G. Szego, Über die Lebesgueschen Konstanten bei den Fourierschen Reihen, Math. Z. 9 (1921) 163-166.

Eric Weisstein's World of Mathematics, Lebesgue Constants

Formula

a(n) = numerator(R(1;n)) = numerator(45*sum(Theta(1,3*k)/(4*k^2-1),k=1..n)), n>=1.

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_] := Numerator[45*Sum[theta[1, 3*k]/(4*k^2-1), {k, 1, n}]]; Table[a[n], {n, 1, 12}] (* Jean-François Alcover, Dec 02 2013 *)

Crossrefs

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

Sequence in context: A273940 A348307 A033998 * A273192 A233210 A173369

Adjacent sequences: A157164 A157165 A157166 * A157168 A157169 A157170

Keyword

nonn,frac,easy

Author

Wolfdieter Lang, Oct 16 2009, Nov 24 2009

Status

approved