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

1,2

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;

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.

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)).

Example

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.

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]];

Prog

(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

Crossrefs

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

Sequence in context: A215328 A107298 A195104 * A257905 A305878 A093924

Adjacent sequences: A338876 A338877 A338878 * A338880 A338881 A338882

Keyword

nonn,frac,tabl

Author

Sanjar Abrarov, Nov 13 2020

Status

approved