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A281500 Reduced denominators of f(n) = (n+1)/(2^(2+n)-2) with A026741(n+1) as numerators.

%I A281500 #35 Feb 12 2017 13:14:29
%S A281500 2,3,14,15,62,63,254,255,1022,1023,4094,4095,16382,16383,65534,65535,
%T A281500 262142,262143,1048574,1048575,4194302,4194303,16777214,16777215,
%U A281500 67108862,67108863,268435454,268435455,1073741822,1073741823,4294967294,4294967295,17179869182,17179869183
%N A281500 Reduced denominators of f(n) = (n+1)/(2^(2+n)-2) with A026741(n+1) as numerators.
%C A281500 f(n) = (n+1)/A000918(n+2) = 1/2, 2/6, 3/14, 4/30, 5/62, 6/126, 7/254, 8/510, 9/1022, 10/2046, 11/4094, 12/8190, ... .
%C A281500 Partial reduction: 1/2, 1/3, 3/14, 2/15, 5/62, 3/63, 7/254, 4/255, 9/1022, 5/1023, 11/4094, 6/4095, ... = A026741(n+1)/a(n).
%C A281500 Full reduction: 1/2, 1/3, 3/14, 2/15, 5/62, 1/21, 7/254, ... = A111701(n+1)/(2, 3, 14, 15, 62, 21, ... )
%C A281500 A164555(n+1)/A027642(n) = 1/2, 1/6, 0, -1/30, 0, 1/42, ... = f(n) * A198631(n)/A006519(n+1) = 1, 1/2, 0, -1/4, 0, 1/2, ... .).
%C A281500 Via f(n), we go from the second fractional Euler numbers to the second Bernoulli numbers.
%C A281500 a(n) mod 10: periodic sequence of length 4: repeat [2, 3, 4, 5].
%C A281500 a(n) differences table:
%C A281500 .  2,   3, 14,  15,  62,   63, 254,  255, ...
%C A281500 .  1,  11,  1,  47,   1,  191,   1,  767, ... see A198693
%C A281500 . 10, -10, 46, -46, 190, -190, 766, -766, ... see A096045, from Bernoulli(2n).
%C A281500 Extension of a(n): a(-2) = -1, a(-1) = 0.
%H A281500 Colin Barker, <a href="/A281500/b281500.txt">Table of n, a(n) for n = 0..1000</a>
%H A281500 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,5,0,-4).
%F A281500 From _Colin Barker_, Jan 24 2017: (Start)
%F A281500 G.f.: (2 + 3*x + 4*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)).
%F A281500 a(n) = 5*a(n-2) - 4*a(n-4) for n>3. (End)
%F A281500 From _Jean-François Alcover_, Jan 24 2017: (Start)
%F A281500 a(n) = (3 + (-1)^n)*(2^(n + 1) - 1)/2.
%F A281500 a(n) = 4^((n + 1 + ((n + 1) mod 2))/2) - 1 - ((n + 1) mod 2). (End)
%F A281500 a(n) = a(n-2) + A117856(n+1) for n>1.
%F A281500 a(2*k) = 4^(k + 1) - 2, a(2*k+1) = a(2*k) + 1 = 4^(k+1) - 1.
%F A281500 a(2*k) + a(2*k+1) = A267921(k+1).
%t A281500 a[n_] := (3+(-1)^n)*(2^(n+1)-1)/2; (* or *) a[n_] := If[EvenQ[n], 4^(n/2+1)-2, 4^((n+1)/2)-1]; Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Jan 24 2017 *)
%o A281500 (PARI) Vec((2 + 3*x + 4*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)) + O(x^50)) \\ _Colin Barker_, Jan 24 2017
%Y A281500 Cf. A000027, A000918, A001477, A006519, A026741, A027642, A096045, A111701, A117856, A164555, A198631, A198693, A209308, A267921.
%K A281500 nonn,easy
%O A281500 0,1
%A A281500 _Paul Curtz_, Jan 23 2017