A338879 - OEIS

%I #67 Dec 23 2020 08:42:39

%S 1,3,2,5,11,3,7,11,9,4,9,37,31,27,5,11,28,8,7,19,6,13,79,69,61,55,51,

%T 7,15,53,47,21,19,35,33,8,17,137,41,37,101,31,29,83,9,19,86,78,71,13,

%U 12,56,53,51,10,21,211,193,177,163,151,141,133,127,123,11,23,127,39,18,50,31,29,41,13,25,73,12

%N Denominators in a set of expansions of the single-term Machin-like formula for Pi.

%C Numerators are A338878.

%C Abrarov et al. give an identity arctan(n*x) = Sum_{m=1..n} arctan(x / (1 + (m-1)*m*x^2)). At x=1/n this identity provides set of expansions of the single-term Machin-like formula for Pi in form Pi/4 = arctan(1) = Sum_{m=1..n} arctan(n/((m-1)*m + n^2)). For m = n - k + 1 at k=1..n the fractions n / ((m-1)*m + n^2) constitute the triangle with rows in ascending order:

%C     k=  1     2     3     4     5     6

%C   n=1:  1;

%C   n=2:  1/3,  1/2;

%C   n=3:  1/5,  3/11, 1/3;

%C   n=4:  1/7,  2/11, 2/9,  1/4;

%C   n=5:  1/9,  5/37, 5/31, 5/27, 1/5;

%C   n=6:  1/11, 3/28, 1/8,  1/7,  3/19, 1/6;

%H Sanjar Abrarov, <a href="/A338879/b338879.txt">Table of n, a(n) for n = 1..120</a>

%H Sanjar M. Abrarov, Rehan Siddiqui, Rajinder K. Jagpal, and Brendan M. Quine, <a href="https://arxiv.org/abs/2004.11711">Unconditional applicability of the Lehmer's measure to the two-term Machin-like formula for pi</a>, arXiv:2004.11711 [math.GM], 2020.

%F T(n,k) = denominator of n / ((n-k)*(n-k+1) + n^2), for n>=1 and 1 <= k <= n.

%F Pi/4 = Sum_{k=1..n} arctan(A338878(n,k) / T(n,k)).

%e The triangle T(n,k) begins:

%e     k=  1   2   3   4   5   6

%e   n=1:  1;

%e   n=2:  3,  2;

%e   n=3:  5,  11, 3;

%e   n=4:  7,  11, 9,  4;

%e   n=5:  9,  37, 31, 27, 5;

%e   n=6:  11, 28, 8,  7,  19, 6;

%e For example, at n = 3 the expansion formula is Pi/4 = arctan(1/5) + arctan(3/11) + arctan(1/3) and the corresponding sequence in the denominators is 5,11,3.

%t (*Define variable*)

%t PiOver4[m_] := Sum[ArcTan[m/((k - 1)*k + m^2)], {k, 1, m}];

%t (*Expansions*)

%t m := 1;

%t While[m <= 10,

%t   If[m == 1, Print["\[Pi]/4 = ArcTan[1/1]"],

%t     Print["\[Pi]/4 = ", PiOver4[m]]]; m = m + 1];

%t (*Verification*)

%t m := 1;

%t While[m <= 10, Print[PiOver4[m] == Pi/4]; m = m + 1];

%t (*Denominators*)

%t For[n = 1, n <= 10, n++, {k := 1; sq := {};

%t   While[n >= k, AppendTo[sq, Denominator[n/((n - k)*(n - k

%t     + 1) + n^2)]]; k++]}; Print[sq]];

%o (PARI) T(n, k) = if (n>=k, denominator(n/((n - k)*(n - k + 1) + n^2)))

%o matrix(10, 10, n, k, T(n, k)) \\ _Michel Marcus_, Nov 14 2020

%Y Cf. A338878 (numerators), A003881 (Pi/4).

%K nonn,frac,tabl

%O 1,2

%A _Sanjar Abrarov_, Nov 13 2020