A309091 Decimal expansion of 4/(Pi-2).

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

Offset: 1

Author

Alois P. Heinz, Jul 11 2019

Keywords

• nonn
• cons

Comments

This can be computed using a recursion formula discovered by an algorithm called "The Ramanujan Machine":

1*3

4/(Pi-2) = 3 + --------------------

2*4

5 + ----------------

3*5

7 + ------------

4*6

9 + --------

11 + ... .

For a proof by humans see the arXiv:1907.00205 preprint linked below.

Examples

			3.50387678776821732240781940302290775850079601361148312728094190...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• Gal Raayoni, George Pisha, Yahel Manor, Uri Mendlovic, Doron Haviv, Yaron Hadad, and Ido Kaminer, The Ramanujan Machine: Automatically Generated Conjectures on Fundamental Constants, arXiv:1907.00205 [cs.LG], 2019-2020.
• The Ramanujan Machine, Using algorithms to discover new mathematics.
• Index entries for transcendental numbers.

Crossrefs

Cf. A000796, A005563, A144396, A309419, A309420.

Programs

Maple

nn:= 126: # number of digits
b:= i-> `if`(i<2*nn, 2*i+1 +i*(i+2)/b(i+1), 1):
evalf(b(1), nn);

Mathematica

RealDigits[4/(Pi-2), 10, 120][[1]] (* Amiram Eldar, Jun 29 2023 *)