A005793 Number of O_1^{2+}(Z)-orbits of Lorentzian modular group. (Formerly M0078)

1, 1, 1, 2, 1, 1, 2, 3, 2, 1, 2, 3, 1, 2, 3, 4, 2, 2, 2, 4, 2, 1, 4, 5, 2, 2, 3, 4, 2, 2, 4, 6, 2, 2, 4, 6, 1, 2, 5, 5, 3, 2, 2, 6, 3, 2, 6, 7, 3, 3, 4, 4, 2, 3, 5, 8, 2, 1, 5, 7, 2, 3, 7, 7, 3, 3, 2, 7, 3, 2, 8, 8, 2, 3, 5, 6, 3, 2, 6, 10, 4, 2, 5, 8, 2, 3, 7, 5, 4, 4, 4, 8, 2, 3, 9, 11, 2, 4, 6, 6, 4, 2, 6, 10

Offset

1,4

Comments

Table 2 of Fox and Parker lists a(n) for n=1 to 1000. - Michael Somos, Jul 13 2013

"Each O_1^{2+}(Z)-orbit has a representative (z, x, y) in Z^3 with z > x >= y >= 0 and z >= x+y" from proof of Corollary 2.4 of Fox and Parker. We are looking for solutions of n = z^2 - x^2 - y^2. - Michael Somos, Jul 13 2013

References

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

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

G. J. Fox, Letter to N. J. A. Sloane, May 1991

Glenn J. Fox and Phillip E. Parker, The Lorentzian modular group and nonlinear lattices, The mathematical heritage of C. F. Gauss, 282-303, World Sci. Publishing, River Edge, NJ, 1991.

Glenn J. Fox and Phillip E. Parker, The Lorentzian modular group and nonlinear lattices II, The mathematical heritage of C. F. Gauss, 282-303, World Sci. Publishing, River Edge, NJ, 1991.

Index entries for sequences related to modular groups

Example

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(8) = 3 since orbits(8) = [[3, 1, 0], [4, 2, 2], [5, 4, 1]] where 8 = 3^2-1^2-0^2 = 4^2-2^2-2^2 = 5^2-4^2-1^2 for the three O_1^{2+}(Z)-orbit representatives. - Michael Somos, Jul 13 2013

Mathematica

r[n_] := Reduce[z > x >= y >= 0 && z >= x + y && n == z^2 - x^2 - y^2, {x, y, z}, Integers]; a[n_] := Switch[rn = r[n]; Head[rn], And, 1, Or, Length[rn]]; Table[Print["a(", n, ") = ", an = a[n], " ", {x, y, z} /. {ToRules[rn]}]; an, {n, 1, 105}]
(* or, from 1st PARI script: *) a[n_] := Sum[If[Mod[n - i, 2] != 0, 0, DivisorSum[j = (n + i*i)/2, Boole[# >= i && #^2 <= j] &]], {i, 0, Sqrt[n] }]; Array[a, 105] (* Jean-François Alcover, Dec 02 2015 *)

Prog

(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 && d*d <= j))))} /* Michael Somos, Feb 14 2002 */
(PARI) {orbits(n) = local(j, v=[]); if( n<1, 0, forstep( i=n%2, sqrtint(n), 2, fordiv( j = (n + i*i) / 2, d, if( d>=i && d*d <= j, v = concat( [[d+j/d-i, j/d-i, d-i]], v)))); vecsort(v))} /* Michael Somos, Jul 13 2013 */

Crossrefs

Cf. A005794.

Sequence in context: A319420 A267134 A098280 * A183391 A029346 A238902

Adjacent sequences: A005790 A005791 A005792 * A005794 A005795 A005796

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Michael Somos, Feb 14 2002

Status

approved