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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 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
KEYWORD
nonn,frac,tabl
AUTHOR
Sanjar Abrarov, Nov 13 2020
STATUS
approved

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Last modified February 24 12:51 EST 2024. Contains 370303 sequences. (Running on oeis4.)