A338878 Numerators in a set of expansions of the single-term Machin-like formula for Pi.

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

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