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
KEYWORD
AUTHOR
Sanjar Abrarov, Nov 13 2020
STATUS
approved