This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A176447 #49 Sep 21 2023 01:45:40
%S A176447 0,1,-1,3,-2,5,-3,7,-4,9,-5,11,-6,13,-7,15,-8,17,-9,19,-10,21,-11,23,
%T A176447 -12,25,-13,27,-14,29,-15,31,-16,33,-17,35,-18,37,-19,39,-20,41,-21,
%U A176447 43,-22,45,-23,47,-24,49,-25,51,-26,53,-27,55,-28,57,-29,59,-30,61,-31,63,-32,65,-33,67,-34,69,-35
%N A176447 a(2n) = -n, a(2n+1) = 2n+1.
%C A176447 There is more complicated way of defining the sequence: consider the sequence of modified Bernoulli numbers EVB(n) = A176327(n)/A176289(n) and its inverse binomial transform IEVB(n) = A176328(n)/A176591(n). Then a(n) is the numerator of the difference EVB(n)-IEVB(n). The denominator of the difference is 1 if n=0, else A040001(n-1).
%C A176447 A particularity of EVB(n) is: its (forward) binomial transform is 1, 1, 7/6, 3/2, 59/30,.. = (-1)^n*IEVB(n).
%C A176447 Note that A026741 is related to the Rydberg-Ritz spectrum of the hydrogen atom.
%H A176447 R. J. Mathar, <a href="/A176447/b176447.txt">Table of n, a(n) for n = 0..1000</a>
%H A176447 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).
%F A176447 From _R. J. Mathar_, Dec 01 2010: (Start)
%F A176447 a(n) = (-1)^n*A026741(n) = n*(1-3*(-1)^n)/4.
%F A176447 G.f.: x*(1-x+x^2) / ( (x-1)^2*(1+x)^2 ).
%F A176447 a(n) = +2*a(n-2) -a(n-4). (End)
%F A176447 a(n) = -a(-n) for all n in Z. - _Michael Somos_, Jun 11 2013
%F A176447 From _Michael Somos_, Aug 30 2014: (Start)
%F A176447 Euler transform of length 6 sequence [ -1, 3, 1, 0, 0, -1].
%F A176447 0 = - 1 - a(n) - a(n+1) + a(n+2) + a(n+3) for all n in Z.
%F A176447 0 = 1 + a(n)*(-2 -a(n) + a(n+2)) - 2*a(n+1) - a(n+2) for all n in Z. (End)
%F A176447 From _Michael Somos_, May 04 2015: (Start)
%F A176447 a(n) is multiplicative with a(2^e) = -(2^(e-1)) if e>0, a(p^e) = p^e otherwise.
%F A176447 G.f.: (f(x) - 3 * f(-x)) / 4 where f(x) := x / (1 - x)^2.
%F A176447 G.f.: x * (1 - x) * (1 - x^6) / ((1 - x^2)^3 * (1 - x^3)). (End)
%F A176447 From _Amiram Eldar_, Sep 21 2023: (Start)
%F A176447 Dirichlet g.f.: zeta(s-1) * (1 - 3/2^s).
%F A176447 Sum_{k=0..n} a(k) = A008795(n-1), for n > 0.
%F A176447 Sum_{k=0..n} a(k) ~ n^2/8. (End)
%e A176447 G.f. = x - x^2 + 3*x^3 - 2*x^4 + 5*x^5 - 3*x^6 + 7*x^7 - 4*x^8 + 9*x^9 - 5*x^10 + ...
%t A176447 a[n_?EvenQ]:=-(n/2); a[n_?OddQ]:=n; Table[a[n], {n, 100}] (* _Alonso del Arte_, Dec 01 2010 *)
%t A176447 a[ n_] := n / If[ Mod[ n, 2] == 1, 1, -2]; (* _Michael Somos_, Jun 11 2013 *)
%t A176447 CoefficientList[Series[x (1 - x + x^2)/((x - 1)^2*(1 + x)^2), {x, 0, 70}], x] (* _Michael De Vlieger_, Dec 10 2016 *)
%t A176447 LinearRecurrence[{0,2,0,-1},{0,1,-1,3},80] (* _Harvey P. Dale_, Nov 01 2017 *)
%o A176447 (Magma) [n*(1-3*(-1)^n)/4: n in [0..60]]; // _Vincenzo Librandi_, Aug 04 2011
%o A176447 (PARI) {a(n) = n / if( n%2, 1, -2)}; /* _Michael Somos_, Jun 11 2013 */
%Y A176447 Cf. A008795, A026741, A040001, A176289, A176327, A176328, A176591.
%K A176447 sign,easy,mult
%O A176447 0,4
%A A176447 _Paul Curtz_, Apr 18 2010