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A005794
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Number of SO_1^{2+}(Z) orbits of Lorentzian modular group.
(Formerly M0079)
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1
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1, 1, 1, 2, 1, 1, 2, 3, 2, 1, 3, 3, 1, 2, 4, 4, 2, 2, 3, 5, 2, 1, 6, 5, 2, 3, 4, 4, 3, 2, 6, 7, 2, 2, 6, 7, 1, 3, 8, 5, 4, 2, 3, 9, 3, 2, 10, 7, 3, 4, 6, 5, 3, 4, 8, 10, 2, 1, 9, 8, 3, 4, 10, 8, 4, 4, 3, 10, 4, 2, 14, 8, 2, 5, 7, 9, 4, 2, 10, 13, 5, 2, 9, 10
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OFFSET
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1,4
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COMMENTS
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Each SO_1^{2+}(Z) orbit has a representative (z, x, y) in Z^3 with z > x >= 0, z > y >= 0 and z >= x+y. We are looking for solutions of n = z^2 - x^2 - y^2. - Michael Somos, Jul 13 2013
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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EXAMPLE
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x + x^2 + x^3 + 2*x^4 + x^5 + x^6 + 2*x^7 + 3*x^8 + 2*x^9 + x^10 + 3*x^11 + ...
a(11) = 3 since orbits(11) = [[4, 1, 2], [4, 2, 1], [6, 5, 0]] where 11 = 4^2-1^2-2^2 = 4^2-2^2-1^2 = 6^2-5^2-0^2 for the three SO_1^{2+}(Z) orbit representatives.
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MATHEMATICA
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a[n_] := Sum[If[Mod[n-i, 2] == 1, 0, j = (n+i*i)/2; DivisorSum[j, Boole[# >= i && j >= #*i && (j <= #^2 || (i>0 && # > i && j > #*i))]&]], {i, 0, Floor[Sqrt[n]]}]; Array[a, 105] (* Jean-François Alcover, Dec 03 2015, adapted from PARI *)
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PROG
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(PARI) {a(n) = my(j); if( n<1, 0, sum( i=0, sqrtint(n), if( (n-i)%2, 0, sumdiv( j = (n + i*i) / 2, d, d>=i && j>=d*i && (j<=d*d || (i>0 && d>i && j>d*i))))))} /* Michael Somos, Jul 13 2013 */
(PARI) {orbits(n) = local(j, v=[], x, y, z); if( n<1, 0, forstep( i=n%2, sqrtint(n), 2, fordiv( j = (n + i*i) / 2, d, x = d-i; y = j/d-i; z = x+y+i; if( x>=0 && y>=0 && (y<=x || (i>0 && x>0 && y>0)), v = concat([[z, y, x]], v)))); vecsort(v))} /* Michael Somos, Jul 13 2013 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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