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%I #42 Apr 23 2023 07:27:33
%S 1,2,3,6,30,30,42,42,30,30,66,66,2730,2730,6,6,510,510,798,798,330,
%T 330,138,138,2730,2730,6,6,870,870,14322,14322,510,510,6,6,1919190,
%U 1919190,6,6,13530,13530,1806,1806,690,690,282,282,46410,46410,66,66,1590,1590
%N 1, followed by denominators of first differences of Bernoulli numbers (B(i)-B(i-1)).
%C Equivalently, denominators of Bernoulli twin numbers C(n) (cf. A051716).
%C 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_.
%C Denominators of column 1 of table described in A051714/A051715.
%H G. C. Greubel, <a href="/A051717/b051717.txt">Table of n, a(n) for n = 0..5000</a>
%H M. Kaneko, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/KANEKO/AT-kaneko.html">The Akiyama-Tanigawa algorithm for Bernoulli numbers</a>, J. Integer Sequences, 3 (2000), #00.2.9.
%e Bernoulli numbers: 1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, ...
%e First differences: -3/2, 2/3, -1/6, -1/30, 1/30, 1/42, -1/42, -1/30, ...
%e Numerators: -3, 2, -1, -1, 1, 1, -1, -1, 1, 5, -5, -691, 691, 7, ...
%e Denominators: 2, 3, 6, 30, 30, 42, 42, 30, 30, 66, 66, 2730, ...
%e 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, ...
%p 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;
%t 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 *)
%t Join[{1},Denominator[Total/@Partition[BernoulliB[Range[0,60]],2,1]]] (* _Harvey P. Dale_, Mar 09 2013 *)
%t Join[{1},Denominator[Differences[BernoulliB[Range[0,60]]]]] (* _Harvey P. Dale_, Jun 28 2021 *)
%o (PARI) a(n)=if(n<3,n+1,denominator(bernfrac(n)-bernfrac(n-1))) \\ _Charles R Greathouse IV_, May 18 2015
%o (Magma)
%o f:= func< n | Bernoulli(n) + Bernoulli(n-1) >;
%o function A051717(n)
%o if n eq 0 then return 1;
%o elif (n mod 2) eq 0 then return Denominator(f(n));
%o else return Denominator(-f(n));
%o end if;
%o end function;
%o [A051717(n): n in [0..50]]; // _G. C. Greubel_, Apr 22 2023
%o (SageMath)
%o def f(n): return bernoulli(n)+bernoulli(n-1)
%o def A051717(n):
%o if (n==0): return 1
%o elif (n%2==0): return denominator(f(n))
%o else: return denominator(-f(n))
%o [A051717(n) for n in range(51)] # _G. C. Greubel_, Apr 22 2023
%Y Cf. A027641, A027642, A051714, A051715, A051716, A129825, A129826.
%Y Cf. A129724.
%Y For numerators see A172083.
%K nonn,easy,nice,frac
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _James A. Sellers_, Dec 08 1999
%E Edited by _N. J. A. Sloane_, May 25 2008
%E Entry revised by _N. J. A. Sloane_, Apr 22 2021
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