login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A005794 Number of SO_1^{2+}(Z) orbits of Lorentzian modular group.
(Formerly M0079)
1

%I M0079

%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-Fran├žois 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_.

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 13 17:17 EST 2019. Contains 329970 sequences. (Running on oeis4.)