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A338878 Numerators in a set of expansions of the single-term Machin-like formula for Pi. 2
1, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 5, 5, 5, 1, 1, 3, 1, 1, 3, 1, 1, 7, 7, 7, 7, 7, 1, 1, 4, 4, 2, 2, 4, 4, 1, 1, 9, 3, 3, 9, 3, 3, 9, 1, 1, 5, 5, 5, 1, 1, 5, 5, 5, 1, 1, 11, 11, 11, 11, 11, 11, 11, 11, 11, 1, 1, 6, 2, 1, 3, 2, 2, 3, 1, 2, 6, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Denominators are A338879.

Abrarov et al. (see section LINKS) 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;

Section EXAMPLE shows the corresponding triangle T(n,k) with numerators. This triangle T(n,k) possesses a sifting property for primes. In particular, in a row of kind {1,p,p,p,...,p,p,1} the integer p must be a prime and n = p except the case n = 4 when prime p = 2.

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) = numerator(n/((n-k)*(n-k+1) + n^2)) = n/gcd(n,k*(k-1)), for n>=1 and 1 <= k <= n.

Pi/4 = Sum_{k=1..n} arctan(T(n,k) / A338879(n,k)).

EXAMPLE

Triangle T(n,k) begins:

     k= 1  2  3  4  5  6

  n=1:  1;

  n=2:  1, 1;

  n=3:  1, 3, 1;

  n=4:  1, 2, 2, 1;

  n=5:  1, 5, 5, 5, 1;

  n=6:  1, 3, 1, 1, 3, 1;

For example, for row n = 3 the corresponding expansion formula is Pi/4 = arctan(1/5) + arctan(3/11) + arctan(1/3) and the numerators are 1,3,1.

At n = 3, n = 4 and n = 5 the rows are {1,3,1}, {1,2,2,1} and {1,5,5,5,1} and the primes are 3, 2 and 5, respectively.

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];

(*Numerators*)

For[n = 1, n <= 15, n++, {k := 1; sq := {};

  While[n >= k, AppendTo[sq, n/GCD[n, k*(k - 1)]]; k++]};

    Print[sq]];

PROG

(PARI) T(n, k) = if (n>=k, n/gcd(n, k*(k - 1)))

matrix(10, 10, n, k, T(n, k)) \\ Michel Marcus, Nov 14 2020

CROSSREFS

Cf. A338879 (denominators), A003881 (Pi/4).

Sequence in context: A153066 A126209 A176346 * A073166 A050169 A143214

Adjacent sequences:  A338875 A338876 A338877 * A338879 A338880 A338881

KEYWORD

nonn,frac,tabl

AUTHOR

Sanjar Abrarov, Nov 13 2020

STATUS

approved

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Last modified June 20 01:09 EDT 2021. Contains 345154 sequences. (Running on oeis4.)