A005794 - OEIS

%I M0079 #29 Oct 10 2019 08:13:35

%S 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,

%T 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,

%U 10,4,2,14,8,2,5,7,9,4,2,10,13,5,2,9,10

%N Number of SO_1^{2+}(Z) orbits of Lorentzian modular group.

%C 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

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H G. J. Fox, <a href="/A005793/a005793.pdf">Letter to N. J. A. Sloane, May 1991</a>

%H Glenn J. Fox and Phillip E. Parker, <a href="https://www.researchgate.net/publication/264911257">The Lorentzian modular group and nonlinear lattices</a>, The mathematical heritage of C. F. Gauss, 282-303, World Sci. Publishing, River Edge, NJ, 1991.

%H Glenn J. Fox and Phillip E. Parker, <a href="https://www.researchgate.net/publication/269409370">The Lorentzian modular group and nonlinear lattices II</a>, The mathematical heritage of C. F. Gauss, 282-303, World Sci. Publishing, River Edge, NJ, 1991.

%H <a href="/index/Gre#groups_modular">Index entries for sequences related to modular groups</a>

%e 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 + ...

%e 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.

%t 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 *)

%o (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 */

%o (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 */

%Y Cf. A005793.

%K nonn

%O 1,4

%A _N. J. A. Sloane_.