A005794 - OEIS

A005794

Number of SO_1^{2+}(Z) orbits of Lorentzian modular group.
(Formerly M0079)

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

1,4

COMMENTS

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

REFERENCES

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

LINKS

Table of n, a(n) for n=1..84.

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(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.

MATHEMATICA

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 *)

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 && 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 */

CROSSREFS

Cf. A005793.

Sequence in context: A238902 A329750 A030496 * A280860 A208993 A328029

Adjacent sequences: A005791 A005792 A005793 * A005795 A005796 A005797

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

STATUS

approved