A013705 Decimal expansion of 4*Sum_{k=1..500000} (-1)^(k-1)/(2k-1).

3, 1, 4, 1, 5, 9, 0, 6, 5, 3, 5, 8, 9, 7, 9, 3, 2, 4, 0, 4, 6, 2, 6, 4, 3, 3, 8, 3, 2, 6, 9, 5, 0, 2, 8, 8, 4, 1, 9, 7, 2, 9, 1, 3, 9, 9, 3, 7, 5, 1, 0, 3, 0, 5, 0, 9, 7, 4, 9, 4, 4, 6, 9, 3, 3, 4, 9, 8, 1, 6, 4, 0, 0, 8, 8, 0, 6, 7, 8, 9, 9, 9, 0, 2, 6, 7, 5, 6, 7, 8, 7, 3, 0, 3, 3, 3, 4, 0, 4, 3, 6, 9, 6, 9, 5

Offset: 1

Author

N. J. A. Sloane

Keywords

• cons
• nonn
• easy

Comments

An approximation to Pi.

A case of "high precision fraud": curiously, among the first 40 digits, only 4 are wrong (in positions 7, 18, 19 and 30). - Jean-François Alcover, Apr 23 2013

This result arises because the sum is Pi - 2*10^-6 + 2*10^-18 - 10^-29 + 122*10^-42 - ... - Jon E. Schoenfield, Mar 11 2018

The constant is rational, as a finite product of rational numbers. The period of its decimal expansion is L = 1.7368897... * 10^33024, and so a(n + L) = a(n) for large enough n.

Examples

			3.1415906535897932404626433832695028841972913993751030509749446933498...
		

Links

• J. M. Borwein and P. B. Borwein, Strange series and high precision fraud, Amer. Math. Monthly 99, 7 (Aug. 1992), 622-640.
• J. M. Borwein, P. B. Borwein and K. Dilcher, Pi, Euler numbers and asymptotic expansions, Amer. Math. Monthly, 96 (1989), 681-687.
• J. M. Borwein and R. M. Corless, Review of "An Encyclopedia of Integer Sequences" by N. J. A. Sloane and Simon Plouffe, SIAM Review, 38 (1996), 333-337.
• Index entries for linear recurrences with constant coefficients.

Crossrefs

Cf. A000796, A216542, A013706, A216543, A216544, A216545, A013705, A216546, A216547, A216548.

Cf. also A195793.

Programs

PARI

4*sum(k=1, 500000, (-1.)^(k-1)/(2*k-1)) \\ Michel Marcus, Mar 11 2018

Extensions

a(78)-a(80) corrected and more digits from Jon E. Schoenfield, Mar 11 2018