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

1,1

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).

Links

Table of n, a(n) for n=1..85.

Formula

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

Example

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).

Sequence in context: A290902 A155117 A137638 * A145839 A232289 A370480

Adjacent sequences: A156016 A156017 A156018 * A156020 A156021 A156022

Keyword

nonn,frac

Author

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

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

Status

approved