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

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

Examples

			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
		

Crossrefs

Cf. A002485, A002486, A156019 (numerators).

Programs

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

Formula

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

Extensions

More terms from Alexander R. Povolotsky, Sep 01 2009

More terms from Michel Marcus, Jan 05 2022