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A051717
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1, followed by denominators of first differences of Bernoulli numbers (B(i)-B(i-1)).
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30
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1, 2, 3, 6, 30, 30, 42, 42, 30, 30, 66, 66, 2730, 2730, 6, 6, 510, 510, 798, 798, 330, 330, 138, 138, 2730, 2730, 6, 6, 870, 870, 14322, 14322, 510, 510, 6, 6, 1919190, 1919190, 6, 6, 13530, 13530, 1806, 1806, 690, 690, 282, 282, 46410, 46410, 66, 66, 1590, 1590
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OFFSET
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0,2
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COMMENTS
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Equivalently, denominators of Bernoulli twin numbers C(n) (cf. A051716).
The Bernoulli twin numbers C(n) are defined by C(0) = 1, then C(2n) = B(2n) + B(2n-1), C(2n+1) = -B(2n+1) - B(2n), where B() are the Bernoulli numbers A027641/A027642. The definition is due to Paul Curtz.
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LINKS
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EXAMPLE
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Bernoulli numbers: 1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, ...
First differences: -3/2, 2/3, -1/6, -1/30, 1/30, 1/42, -1/42, -1/30, ...
Numerators: -3, 2, -1, -1, 1, 1, -1, -1, 1, 5, -5, -691, 691, 7, ...
Denominators: 2, 3, 6, 30, 30, 42, 42, 30, 30, 66, 66, 2730, ...
Sequence of C(n)'s begins: 1, -1/2, -1/3, -1/6, -1/30, 1/30, 1/42, -1/42, -1/30, 1/30, 5/66, -5/66, -691/2730, 691/2730, 7/6, -7/6, ...
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MAPLE
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C:=proc(n) if n=0 then RETURN(1); fi; if n mod 2 = 0 then RETURN(bernoulli(n)+bernoulli(n-1)); else RETURN(-bernoulli(n)-bernoulli(n-1)); fi; end;
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MATHEMATICA
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c[0]= 1; c[n_?EvenQ]:= BernoulliB[n] + BernoulliB[n-1]; c[n_?OddQ]:= -BernoulliB[n] - BernoulliB[n-1]; Table[Denominator[c[n]], {n, 0, 53}] (* Jean-François Alcover, Dec 19 2011 *)
Join[{1}, Denominator[Total/@Partition[BernoulliB[Range[0, 60]], 2, 1]]] (* Harvey P. Dale, Mar 09 2013 *)
Join[{1}, Denominator[Differences[BernoulliB[Range[0, 60]]]]] (* Harvey P. Dale, Jun 28 2021 *)
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PROG
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(Magma)
f:= func< n | Bernoulli(n) + Bernoulli(n-1) >;
if n eq 0 then return 1;
elif (n mod 2) eq 0 then return Denominator(f(n));
else return Denominator(-f(n));
end if;
end function;
(SageMath)
def f(n): return bernoulli(n)+bernoulli(n-1)
if (n==0): return 1
elif (n%2==0): return denominator(f(n))
else: return denominator(-f(n))
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CROSSREFS
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KEYWORD
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nonn,easy,nice,frac
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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