VOOZH about

URL: https://oeis.org/A319150

⇱ A319150 - OEIS

login
A319150
a(n) = gcd(A275286(n), A001818(n+1)) / A025549(n+1)^2.
0
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
OFFSET
0,28
COMMENTS
A(n) = a(n)*A025549(n+1)^2 = gcd(B(n), C(n)).
B(n) = A275286(n).
C(n) = A001818(n+1).
At first gcd(B(n), C(n)) = A025549(n+1)^2, but from n = 27 to n = 37, gcd(B(n), C(n)) = 11*A025549(n+1)^2, and then comes back to normal, then equals 19*A025549(n+1)^2, comes back to normal again, and so on ...
Let S(n) = Sum_{k=0..n} ((-1)^k)/(2*k+1)^2 (S(n) is NOT an integer sequence).
Notice that when n approaches +oo, D(n) converges to Catalan's constant (A006752).
A294970(n) is equal to the numerator of S(n) (when reduced).
Therefore B(n)/A(n) = A294970(n)
A294971(n) is equal to the denominator of S(n) (when reduced).
Therefore C(n)/A(n) = A294971(n).
This sequence was used to study the expression B(n)/C(n) (which equals S(n)) in an attempt to find out if Catalan's constant is irrational.
FORMULA
Explicit formula:
a(n) = gcd( ((2*n+1)!!)^2 * (Sum_{i=0..n}((-1)^i)/(2*i+1)^2), ((2*n+1)!!)^2 ) / ( (((2*n+1)!!)^2) / ( lcm{1,3,5,...,2*n+1} ) )^2.
A few relations:
gcd(A275286(n), A001818(n+1)) = a(n)*A025549(n+1)^2 = A(n);
A275286(n)/A(n) = A294970(n);
A001818(n+1)/A(n) = A294971(n);
Limit_{n->+oo} A294970(n)/A294971(n) = G (Catalan's Constant, decimal expansion: A006752).
EXAMPLE
For n = 5:
B(n) = A275286(5) = 98607816;
C(n) = A001818(5+1) = 108056025;
gcd(98607816,108056025) = A(5) = 9;
A025549(5+1)^2 = 3^2 = 9;
So a(5) = A(5)/A025549(5+1)^2 = 9/9 = 1.
MATHEMATICA
a[n_] := GCD[(2n+1)!!^2 * Sum[(-1)^k/(2k+1)^2, {k, 0, n}], (2n+1)!!^2]*
LCM @@ Range[1, 2n+1, 2]^2 / ((2n+1)!!)^2; Array[a, 100, 0] (* Amiram Eldar, Nov 16 2018 *)
PROG
(PARI) dfo(n) = (2*n)! / n! / 2^n;
a6(n) = dfo(n+1)^2*sum(k=0, n, (-1)^k/(2*k+1)^2);
a8(n) = ((2*n)!/(n!*2^n))^2;
a9(n) = (((2*n)!/n!)/2^n)/lcm(vector(n, i, 2*i-1));
a(n) = gcd(a6(n) , a8(n+1)) / a9(n+1)^2; \\ Michel Marcus, Nov 08 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Tristan Cam, Nov 08 2018
STATUS
approved