0,2

I am in the process of editing this entry. The old terms were as follows:

446685,187698,45123,18960,4557,1914,459,192,45,18,3,0,3,6,21,48,195,

462,1917,4560,18963,45126,446688

-446685,-187698,-45123,-18960,-4557,-1914,-459,-192,-45,-18,-3,0,3,6,21,48,195,462,

1917,4560,18963,45126,446688

Michael Somos tells me he will submit the negative terms as a new sequence, A138976. - N. J. A. Sloane (njas(AT)research.att.com), Apr 19 2008

Perfect square values for discriminants are used to classify the Galois group of a polynomial. The O+ discriminant component is Sqrt[6(x^2-3x+6)] (used to generate these values) and for the O- discriminant Sqrt[6(x^2+3x+6)].

This sequence is the negative of the O+ sequence. Also, note that if 3*a[n] represents the positive terms, the negative terms are generated from 3 - 3*a[n].

For the O- sequence reverse the O+ sequence and change all of the signs to generate ...-446688, -45126, -18963, -4560,-1917,-462,-195,-48,-21,-6,-3,0,3,18,45,192,459,1914,4557,18960,45123,187698,446685.

Note that the difference equation a[n] generates the above sequence divided by 3 or ...,-148895, -62566, -15041, -6320, -1519, -638, -153, -64, -15, -6, -1, 0, 1, 2, 7, 16, 65, 154, 639, 1520, 6321, 15042, 148896,...

This sequence, its reverse and the division by 3 form, all appear to be new.

The physics reference is G. W. King, "The Asymmetric Rotor I. Calculation and Symmetry Classification of Energy Levels", Journal of Chemical Physics, Jan 1943, Volume 11, p27-42.

The difference equation is a[n]=11(a[n-2] - a[n-4])+a[n-6] with a[0]=0, a[1]=1, a[2]=2, a[3]=7, a[4]=16, a[5]=65. The solution is for even n: a[n]=(1/2) - (1/12)*(3+2*Sqrt[6])*(5-2*Sqrt[6])^(n/2)+(1/12)*(-3+2*Sqrt[6])*(5+2*Sqrt[6])^(n/2), for odd n a[n]=(1/2) - (1/12)*(3*Sqrt[2]+Sqrt[3])*(5-2*Sqrt[6])^(n/2)+(1/12)*(3*Sqrt[2]-Sqrt[3])*(5+2*Sqrt[6])^(n/2). Multiply the resultant sequence by 3 to generate the present sequence.

G.f.: 3 * (x + x^2 - 5*x^3 - x^4) / (1 - x - 10*x^2 + 10*x^3 + x^4 - x^5). - Michael Somos Apr 05 2008

Do[If[IntegerQ[Sqrt[6 (6 - 3 x + x^2)]], Print[{x, Sqrt[6 (6 - 3 x + x^2)]}]], {x, -1000, 1000}]; Do[If[IntegerQ[Sqrt[6 (6 + 3 x + x^2)]], Print[{x, Sqrt[6 (6 + 3 x + x^2)]}]], {x, -1000, 1000}];

(PARI) {a(n) = local(m); m = if( n<0, m = 1-n, n); (n<0) + (-1)^(n<0) * polcoeff( 3 * (x + x^2 - 5*x^3 - x^4) / ((1 - x) * (1 - 10*x^2 + x^4)) + x*O(x^m), m)} /* Michael Somos Apr 05 2008 */

A138976(n) = a(-n). Also 3*A129444(n+1) = a(n).

Sequence in context: A076102 A094282 A124493 this_sequence A063683 A151396 A148604

Adjacent sequences: A136328 A136329 A136330 this_sequence A136332 A136333 A136334

nonn,uned

Lorenz H. Menke, Jr. (lnz2004(AT)mindspring.com), Mar 27 2008

I rather feel that this should be broken up into two sequences, one each for the positive and negative terms, both starting at 0. - N. J. A. Sloane (njas(AT)research.att.com), Apr 04 2008

More terms from Michael Somos, Apr 05 2008