A156019 Numerators in an infinite sum for Pi.

3, 15, 73, 1, 2, 3, 7, 1, 2, 2, 1, 2, 1, 3, 1, 2, 6, 1, 1, 3, 1, 6, 1, 1, 3, 1, 1, 3, 1, 3, 1, 1, 2, 6, 1, 2, 3, 1, 1, 1, 45, 22, 2, 1, 1, 24, 2, 1, 2, 1, 2, 4, 2, 8, 5, 1, 1, 1, 2, 7, 1, 3, 1, 7, 4, 7, 3, 3, 9, 9, 1, 18, 3, 15, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1

Offset: 1

Author

Gary W. Adamson and Alexander R. Povolotsky, Feb 01 2009

Keywords

• nonn
• frac

Comments

For k >= 0, define Q(k) = A002485(2k)/A002486(2k) (convergents to Pi that are less than Pi), so Pi = Sum_{k>=1} (Q(k) - Q(k-1)). Then a(n) is the numerator of Q(n) - Q(n-1).

Examples

			a(2) = 15 since A002485(4)/A002486(4) = 333/106, A002485(2)/A002486(2) = 3/1, and 333/106 - 3/1 = 15/106 (see table below).
Pi = 3/1 + 15/106 + 73/877203 + 1/2195225334 + 2/17599271777 + 3/360950005720 + 7/17348726394920 + ....
.
  n  Q(n) = A002485(2n)/A002486(2n)  Q(n) - Q(n-1)  a(n)
  -  ------------------------------  -------------  ----
  0       0/1     = 0                     -           -
  1       3/1     = 3                    3/1          3
  2     333/106   = 3.1415094339...     15/106       15
  3  103993/33102 = 3.1415926530...     73/877203    73
		

Crossrefs

Cf. A000796, A002485, A002486, A156020 (denominators).

Formula

a(n) = numerator(A002485(2n)/A002486(2n) - A002485(2n-2)/A002486(2n-2)).

Extensions

More terms from Alexander R. Povolotsky, Sep 01 2009

Edited by Jon E. Schoenfield, Jan 04 2022

More terms from Jinyuan Wang, Jun 29 2022