A338879 Denominators in a set of expansions of the single-term Machin-like formula for Pi.

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

Offset: 1

Author

Sanjar Abrarov, Nov 13 2020

Keywords

• nonn
• frac
• tabl

Comments

Numerators are A338878.

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:

k= 1 2 3 4 5 6

n=1: 1;

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

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

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

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

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

Examples

			The triangle T(n,k) begins:
    k=  1   2   3   4   5   6
  n=1:  1;
  n=2:  3,  2;
  n=3:  5,  11, 3;
  n=4:  7,  11, 9,  4;
  n=5:  9,  37, 31, 27, 5;
  n=6:  11, 28, 8,  7,  19, 6;
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.
		

Links

• Sanjar Abrarov, Table of n, a(n) for n = 1..120
• Sanjar M. Abrarov, Rehan Siddiqui, Rajinder K. Jagpal, and Brendan M. Quine, Unconditional applicability of the Lehmer's measure to the two-term Machin-like formula for pi, arXiv:2004.11711 [math.GM], 2020.

Crossrefs

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

Programs

Mathematica

(*Define variable*)
PiOver4[m_] := Sum[ArcTan[m/((k - 1)*k + m^2)], {k, 1, m}];
(*Expansions*)
m := 1;
While[m <= 10,
  If[m == 1, Print["\[Pi]/4 = ArcTan[1/1]"],
    Print["\[Pi]/4 = ", PiOver4[m]]]; m = m + 1];
(*Verification*)
m := 1;
While[m <= 10, Print[PiOver4[m] == Pi/4]; m = m + 1];
(*Denominators*)
For[n = 1, n <= 10, n++, {k := 1; sq := {};
  While[n >= k, AppendTo[sq, Denominator[n/((n - k)*(n - k
    + 1) + n^2)]]; k++]}; Print[sq]];

PARI

T(n, k) = if (n>=k, denominator(n/((n - k)*(n - k + 1) + n^2)))
matrix(10, 10, n, k, T(n, k)) \\ Michel Marcus, Nov 14 2020

Formula

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

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