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Perfect square values for discriminants are used to classify the Galois group of a polynomial. The O+ discriminant component is Sqrt[sqrt(6*(x^2-3x3*x+6)] ) (used to generate these values) and for the O- discriminant Sqrt[sqrt(6*(x^2+3x3*x+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, p27pp. 27-42.
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[sqrt(6]))*(5-2*Sqrt[sqrt(6]))^(n/2) + (1/12)*(-3+2*Sqrt[sqrt(6]))*(5+2*Sqrt[sqrt(6]))^(n/2), for odd even n , a[(n]) =( 1/2) - (1/12)*(3*Sqrt[sqrt(2]) +Sqrt[ sqrt(3]))*(5-2*Sqrt[sqrt(6]))^(n/2) + (1/12)*(3*Sqrt[sqrt(2]) -Sqrt[ sqrt(3]))*(5+2*Sqrt[sqrt(6]))^(n/2) for odd n. Multiply the resultant sequence by 3 to generate the present sequence.
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(PARI) {a(n) = localmy(m); m = if( n<0, m = 1-n, n); 3*(n<0) + 3*(-1)^(n<0) * polcoeff( (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 */
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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, Apr 19 2008
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
G.f. = 3*x + 6*x^2 + 21*x^3 + 48*x^4 + 195*x^5 + 462*x^6 + 1917*x^7 + ...
(PARI) {a(n) = local(m); m = if( n<0, m = 1-n, n); 3*(n<0) + 3*(-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 */
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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, ...
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