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A338879 Denominators in a set of expansions of the single-term Machin-like formula for Pi. 2
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified May 12 17:59 EDT 2021. Contains 343829 sequences. (Running on oeis4.)