|
| |
|
|
A142970
|
|
Numerators of n-th approximants of a continued fraction for Pi-3.
|
|
3
|
|
|
|
0, 1, 6, 61, 660, 8901, 133266, 2303865, 43808040, 928665225, 21386693790, 537861526965, 14540730176700, 423407835413325, 13140639311294250, 434929825450371825, 15237733330856005200, 565064979900590948625, 22056613209702152061750, 905913636742121921038125
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
The corresponding denominators are A001879(n), n>=0.
Pi= 3 + 1^2/(6+3^2/(6+5^2/(6+... See the J.-P. Delahaye reference. R. Rosenthal mentioned this continued fraction in an e-mail to the author Jul 16 2008.
For the approximants in lowest terms cf. the ones for 3*(Pi-3) given by A130411(n)/A130412(n) in lowest terms.
|
|
|
REFERENCES
|
J.-P. Delahaye, Le fascinant nombre pi, Pour la Science, Paris 1997. In German: Pi - die Story, Birkhaeuser, 1999 Basel, p. 87.
|
|
|
LINKS
|
Table of n, a(n) for n=0..19.
W. Lang, Approximants for Pi-3 and more.
|
|
|
FORMULA
|
a(n)= 6*a(n-1) + ((2*n-1)^2)*a(n-2), a(0)=0, a(1)=1.
E.g.f.: (-3*(1+x-sqrt(1-4*x^2))+ 2*(1+x)*arcsin(2*x))/(1-2*x)^(5/2) from the solution of the linear second order differential equation (1-4*x^2)*y''(x) - 2*(8*x+3)*y'(x) - 9*y(x)=0, obtained from the recurrence, with inputs y(0)=0 and y'(0)=1. A special solution is the e.g.f. of the denominators A001879: (1+x)/(1-2*x)^(5/2).
|
|
|
EXAMPLE
|
Approximants a(n)/A001879(n) (not in lowest terms): [0/1]; [1/6]; [6/45]; [61/420]; [660/4725]; [8901/62370];..
Approximants in lowest terms: [0/1]; [1/6]; [2/15]; [61/420]; [44/315]; [989/6930]; ...
|
|
|
CROSSREFS
|
Sequence in context: A152281 A053462 A160751 * A034659 A064088 A191803
Adjacent sequences: A142967 A142968 A142969 * A142971 A142972 A142973
|
|
|
KEYWORD
|
nonn,easy,frac,cofr
|
|
|
AUTHOR
|
Wolfdieter Lang Sep 15 2008
|
|
|
STATUS
|
approved
|
| |
|
|
