%I #30 Aug 05 2024 09:55:11

%S 1,1,-3,-2,5,3,-7,-4,9,5,-11,-6,13,7,-15,-8,17,9,-19,-10,21,11,-23,

%T -12,25,13,-27,-14,29,15,-31,-16,33,17,-35,-18,37,19,-39,-20,41,21,

%U -43,-22,45,23,-47,-24,49,25,-51,-26,53,27,-55,-28,57,29,-59

%N Discriminant of lattice A_n of determinant n+1.

%D J. H. Conway, The Sensual Quadratic Form, Mathematical Association of America, 1997, p. 4.

%D G. L. Watson, Integral Quadratic Forms, Cambridge University Press, p. 2.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,-2,0,-1).

%F a(2n) = (-1)^n*(2*n+1), a(2n+1) = (-1)^n*(n+1). Or (apart from signs and with offset 1), a(n) = n, n odd; n/2, n even.

%F G.f.: (1+x-x^2)/(1+x^2)^2. - _Len Smiley_

%F a(-2-n) = (-1)^n * a(n). - _Michael Somos_, Jun 15 2005

%F a(n) = -2*a(n-2) - a(n-4); a(0)=1, a(1)=1, a(2)=-3, a(3)=-2. - _Harvey P. Dale_, Dec 02 2011

%F a(n) = (-1)^floor(n/2)*A026741(n+1).

%F a(2*n) = A157142(n). a(2*n - 1) = A181983(n). - _Michael Somos_, Feb 22 2016

%e G.f. = 1 + x - 3*x^2 - 2*x^3 + 5*x^4 + 3*x^5 - 7*x^6 - 4*x^7 + 8*x^9 + 5*x^10 + ...

%t CoefficientList[Series[(1+x-x^2)/(1+x^2)^2,{x,0,60}],x] (* or *) LinearRecurrence[{0,-2,0,-1},{1,1,-3,-2},70]

%t a[ n_] := With[{m = n + 1}, m I^m / If[ Mod[ m, 2] == 1, I, -2]]; (* _Michael Somos_, Jun 11 2013 *)

%o (PARI) {a(n) = if( n==-1, 0, (-1)^(n\2) * (n+1) / gcd(n+1, 2))}; /* _Michael Somos_, Jun 15 2005 */

%o (Python)

%o def A030640(n): return (-(n+1>>1) if n&2 else n+1>>1) if n&1 else (-n-1 if n&2 else n+1) # _Chai Wah Wu_, Aug 05 2024

%Y Cf. A026741 is unsigned version.

%Y Cf. A157142, A181983.

%K sign,easy,nice

%O 0,3

%A _N. J. A. Sloane_