A142970 Numerators of n-th approximants of a continued fraction for Pi-3.

0, 1, 6, 61, 660, 8901, 133266, 2303865, 43808040, 928665225, 21386693790, 537861526965, 14540730176700, 423407835413325, 13140639311294250, 434929825450371825, 15237733330856005200, 565064979900590948625, 22056613209702152061750, 905913636742121921038125

Offset: 0

Author

Wolfdieter Lang, Sep 15 2008

Keywords

• nonn
• easy
• frac
• cofr

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.

The above continued fraction for Pi is the particular case n = 0, x = 3 of a result of Ramanujan, previously given by Euler - see Berndt et al., Chapter 12, Entry 25, p. 268. - Peter Bala, Feb 19 2015

Examples

			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]; ...
		

References

• J.-P. Delahaye, Le fascinant nombre pi, Pour la Science, Paris 1997. In German: Pi - die Story, Birkhäuser, 1999 Basel, p. 87.

Links

• B. C. Berndt, R. L. Lamphere, and B. M. Wilson Chapter 12 of Ramanujan's second notebook: Continued fractions, Rocky Mountain Journal of Mathematics, Volume 15, Number 2 (1985), 235-310
• L. Euler, De fractionibus continuis observationes, The Euler Archive, Index Number 123, Section 67.
• Wolfdieter Lang, Approximants for Pi-3 and more
• L. J. Lange, An Elegant Continued Fraction for π, The American Mathematical Monthly, 106 (1999), 456-458.

Crossrefs

Cf. A000796, A001879, A130411, A130412, A254795.

Programs

Magma

I:=[1,6]; [0] cat [n le 2 select I[n] else 6*Self(n-1)+(2*n-1)^2*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Feb 20 2015

Mathematica

RecurrenceTable[{a[0]==0, a[1]==1, a[n]==6 a[n-1] + (2 n-1)^2 a[n-2]}, a, {n, 30}] (* Vincenzo Librandi, Feb 20 2015 *)

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).

a(n) ~ (Pi-3) * 2^(n+3/2) * n^(n+2) / exp(n). - Vaclav Kotesovec, Oct 05 2013