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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
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OFFSET
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0,28
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COMMENTS
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A(n) = a(n)*A025549(n+1)^2 = gcd(B(n), C(n)).
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).
A294971(n) is equal to the denominator of S(n) (when reduced).
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.
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LINKS
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FORMULA
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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:
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EXAMPLE
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For n = 5:
gcd(98607816,108056025) = A(5) = 9;
So a(5) = A(5)/A025549(5+1)^2 = 9/9 = 1.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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