cp's OEIS Frontend

A156020 Denominators in an infinite sum for Pi.

%I A156020 #19 Jan 05 2022 10:43:48
%S A156020 1,106,877203,2195225334,17599271777,360950005720,17348726394920,
%T A156020 1996375977735378,26627865341803449,668044491303666717,
%U A156020 13157161331655387213,7653283960850915182425,3256741424583567733172850,388712386741794886666062286,266182386623377135274423955447
%N A156020 Denominators in an infinite sum for Pi.
%C A156020 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).
%F A156020 a(n) = denominator(A002485(2n)/A002486(2n) - A002485(2n-2)/A002486(2n-2)).
%e A156020 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).
%e A156020 Pi = 3/1 + 15/106 + 73/877203 + 1/2195225334 + 2/17599271777 + 3/360950005720 + 7/17348726394920 + ....
%e A156020 .
%e A156020   n  Q(n) = A002485(2n)/A002486(2n)  Q(n) - Q(n-1)    a(n)
%e A156020   -  ------------------------------  -------------  ------
%e A156020   0       0/1     = 0                     -              -
%e A156020   1       3/1     = 3                    3/1             1
%e A156020   2     333/106   = 3.1415094339...     15/106         106
%e A156020   3  103993/33102 = 3.1415926530...     73/877203   877203
%o A156020 (PARI) cfPi=contfrac(Pi);
%o A156020 vA002485 = concat(1, contfracpnqn(cfPi, #cfPi)[1, ]);
%o A156020 A002485(n) = vA002485[n];
%o A156020 A002486(n) = contfracpnqn(vecextract(cfPi, 2^n-1))[2, 2];
%o A156020 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
%Y A156020 Cf. A002485, A002486, A156019 (numerators).
%K A156020 nonn,frac
%O A156020 1,2
%A A156020 _Gary W. Adamson_ and _Alexander R. Povolotsky_, Feb 01 2009
%E A156020 More terms from _Alexander R. Povolotsky_, Sep 01 2009
%E A156020 More terms from _Michel Marcus_, Jan 05 2022