A136331 - OEIS

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A136331

The discriminant of the characteristic polynomial of the O+ and O- submatrix for spin 3 of the nuclear electric quadrupole Hamiltonian is a perfect square for these values.

0, 3, 6, 21, 48, 195, 462, 1917, 4560, 18963, 45126, 187701, 446688, 1858035, 4421742, 18392637, 43770720, 182068323, 433285446, 1802290581, 4289083728, 17840837475, 42457551822, 176606084157, 420286434480, 1748220004083 (list; graph; refs; listen; history; internal format)

	OFFSET	`0,2`
	COMMENTS	`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 3a[n] represents the positive terms, the negative terms are generated from 3 - 3a[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.`
	REFERENCES	`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.`
	FORMULA	`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+2Sqrt[6])(5-2Sqrt[6])^(n/2)+(1/12)(-3+2Sqrt[6])(5+2Sqrt[6])^(n/2), for odd n a[n]=(1/2) - (1/12)(3Sqrt[2]+Sqrt[3])(5-2Sqrt[6])^(n/2)+(1/12)(3Sqrt[2]-Sqrt[3])(5+2Sqrt[6])^(n/2). Multiply the resultant sequence by 3 to generate the present sequence.` `G.f.: 3 * (x + x^2 - 5x^3 - x^4) / (1 - x - 10x^2 + 10*x^3 + x^4 - x^5). - Michael Somos Apr 05 2008`
	MATHEMATICA	`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}];`
	PROG	`(PARI) {a(n) = local(m); m = if( n<0, m = 1-n, n); (n<0) + (-1)^(n<0) * polcoeff( 3 * (x + x^2 - 5x^3 - x^4) / ((1 - x) (1 - 10x^2 + x^4)) + xO(x^m), m)} /* Michael Somos Apr 05 2008 */`
	CROSSREFS	`A138976(n) = a(-n). Also 3A129444(n+1) = a(n).` `Sequence in context: A076102 A094282 A124493 A063683 A200380 A151396` `Adjacent sequences: A136328 A136329 A136330 * A136332 A136333 A136334`
	KEYWORD	`nonn,uned`
	AUTHOR	`Lorenz H. Menke, Jr. (lnz2004(AT)mindspring.com), Mar 27 2008`
	EXTENSIONS	`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`

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Last modified February 14 17:10 EST 2012. Contains 205644 sequences.