A030640 Discriminant of lattice A_n of determinant n+1.

1, 1, -3, -2, 5, 3, -7, -4, 9, 5, -11, -6, 13, 7, -15, -8, 17, 9, -19, -10, 21, 11, -23, -12, 25, 13, -27, -14, 29, 15, -31, -16, 33, 17, -35, -18, 37, 19, -39, -20, 41, 21, -43, -22, 45, 23, -47, -24, 49, 25, -51, -26, 53, 27, -55, -28, 57, 29, -59

Offset

0,3

References

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

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

Links

Table of n, a(n) for n=0..58.

Index entries for linear recurrences with constant coefficients, signature (0,-2,0,-1).

Formula

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.

G.f.: (1+x-x^2)/(1+x^2)^2. - Len Smiley

a(-2-n) = (-1)^n * a(n). - Michael Somos, Jun 15 2005

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

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

a(2*n) = A157142(n). a(2*n - 1) = A181983(n). - Michael Somos, Feb 22 2016

Example

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 + ...

Mathematica

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

Prog

(PARI) {a(n) = if( n==-1, 0, (-1)^(n\2) * (n+1) / gcd(n+1, 2))}; /* Michael Somos, Jun 15 2005 */
(Python)
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

Crossrefs

Cf. A026741 is unsigned version.

Cf. A157142, A181983.

Sequence in context: A367728 A194748 A323462 * A176447 A145051 A026741

Adjacent sequences: A030637 A030638 A030639 * A030641 A030642 A030643

Keyword

sign,easy,nice

Author

N. J. A. Sloane

Status

approved