login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A153073 Numerators of the convergents of the continued fraction for L(3, chi4), where L(s, chi4) is the Dirichlet L-function for the non-principal character modulo 4. 3
0, 1, 0, 1, 31, 125, 156, 2933, 61749, 64682, 126431, 317544, 443975, 1205494, 1649469, 6153901, 38572875, 121872526, 3451003603, 3572876129, 14169631990, 31912140109, 46081772099, 124075684307, 2651671142546 (list; graph; refs; listen; history; text; internal format)
OFFSET

-2,5

REFERENCES

Leonhard Euler, "Introductio in Analysin Infinitorum", First Part, Articles 175, 284 and 287.

Bruce C. Berndt, "Ramanujan's Notebooks, Part II", Springer-Verlag, 1989. See page 293, Entry 25 (iii).

LINKS

Table of n, a(n) for n=-2..22.

FORMULA

chi4(k) = Kronecker(-4, k); chi4(k) is 0, 1, 0, -1 when k reduced modulo 4 is 0, 1, 2, 3, respectively; chi4 is A101455.

Series: L(3, chi4) = Sum_{k>=1} chi4(k) k^{-3} = 1 - 1/3^3 + 1/5^3 - 1/7^3 + 1/9^3 - 1/11^3 + 1/13^3 - 1/15^3 + ...

Series: L(3, chi4) = Sum_{k>=0} tanh((2k+1) pi/2)/(2k+1)^3. [Ramanujan; see Berndt, page 293]

Closed form: L(3, chi4) = Pi^3/32.

EXAMPLE

L(3, chi4) = 0.9689461462593693804836348458469186... = [0; 1, 31, 4, 1, 18, 21, 1, 1, 2, 1, 2, 1, 3, 6, 3, 28, ...], the convergents of which are 0/1, 1/0, [0/1], 1/1, 31/32, 125/129, 156/161, 2933/3027, 61749/63728, 64682/66755, 126431/130483, 317544/327721, 443975/458204, ..., with brackets marking index 0. Those prior to index 0 are for initializing the recurrence.

MATHEMATICA

nmax = 100; cfrac = ContinuedFraction[Pi^3/32, nmax + 1]; Join[ {0, 1}, Numerator[ Table[ FromContinuedFraction[ Take[cfrac, j] ], {j, 1, nmax + 1} ] ] ]

CROSSREFS

Cf. A153071, A153072, A153074.

Sequence in context: A068021 A131992 A042884 * A042886 A042888 A183836

Adjacent sequences:  A153070 A153071 A153072 * A153074 A153075 A153076

KEYWORD

nonn,frac,easy,changed

AUTHOR

Stuart Clary, Dec 17 2008

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 11 06:55 EST 2016. Contains 279043 sequences.