A156020 Denominators in an infinite sum for Pi.

1, 106, 877203, 2195225334, 17599271777, 360950005720, 17348726394920, 1996375977735378, 26627865341803449, 668044491303666717, 13157161331655387213, 7653283960850915182425, 3256741424583567733172850, 388712386741794886666062286, 266182386623377135274423955447

Offset

1,2

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 denominator of Q(n) - Q(n-1).

Links

Table of n, a(n) for n=1..15.

Formula

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

Example

a(2) = 106 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             1
  2     333/106   = 3.1415094339...     15/106         106
  3  103993/33102 = 3.1415926530...     73/877203   877203

Prog

(PARI) cfPi=contfrac(Pi);
vA002485 = concat(1, contfracpnqn(cfPi, #cfPi)[1, ]);
A002485(n) = vA002485[n];
A002486(n) = contfracpnqn(vecextract(cfPi, 2^n-1))[2, 2];
a(n) = if (n==1, 1, denominator(A002485(2*n)/A002486(2*n) - A002485(2*n-2)/A002486(2*n-2))); \\ Michel Marcus, Jan 05 2022

Crossrefs

Cf. A002485, A002486, A156019 (numerators).

Sequence in context: A253622 A184208 A082177 * A160487 A178546 A096712

Adjacent sequences: A156017 A156018 A156019 * A156021 A156022 A156023

Keyword

nonn,frac

Author

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

Extensions

More terms from Alexander R. Povolotsky, Sep 01 2009

More terms from Michel Marcus, Jan 05 2022

Status

approved