A395002 Number of index 4*n subgroups of SL_2(Z) not containing the negative identity matrix.

1, 8, 22, 32, 205, 608, 1414, 8192, 26122, 78848, 435292, 1490432, 5297149, 28633088, 104775430, 416448512, 2237912062, 8699445248, 37537384276, 202331389952, 830599212946, 3819834441728, 20761206768412, 89520879828992, 433503401086900, 2383252716781568

Offset

1,2

Comments

Recall that SL_2(Z) is generated by the matrices S and T, where S = [[0, -1], [1, 0]] and T = [[1, 1], [0, 1]] (read by columns); and S^2 = (ST)^3 = -I.

The number of subgroups for 4*n+{1,2,3} is 0, hence sequence is only for 4*n.

Links

Paolo Xausa, Table of n, a(n) for n = 1..949

Wilfried Imrich, On the number of subgroups of given index in SL_2(Z), Archiv der Mathematik, vol. 31, 1978, pp. 224-231.

Formula

a(n) = 2*a(n-2) + 4*n*a(n-3) + Sum_{i = 1 .. n-4} a(i)*a(n-3-i) (Theorem 1 from Imrich).

a(n) ~ sqrt((8/3)*Pi)*exp((n/3)*log(4*n)-n/3+(n/2)^(1/3)+(1/2)*log(n)) (Theorem 5 from Imrich).

Maple

a := proc(n) option remember: local i: if n < 4 then [1, 8, 22][n]:
else 2*a(n-2)+4*n*a(n-3)+add(a(i)*a(n-3-i), i = 1 .. n-4): fi: end:
seq(a(n), n = 1 .. 26);

Mathematica

A395002[n_] := A395002[n] = If[n <= 3, {1, 8, 22}[[n]], 2*A395002[n-2] + 4*n*A395002[n-3] + Sum[A395002[i]*A395002[n-3-i], {i, n-4}]];
Array[A395002, 30] (* Paolo Xausa, Apr 22 2026 *)

Crossrefs

Cf. A005133.

Sequence in context: A064193 A301482 A213017 * A305515 A360510 A063301

Adjacent sequences: A394999 A395000 A395001 * A395003 A395004 A395005

Keyword

nonn

Author

Lorenzo Sauras Altuzarra, Apr 09 2026

Status

approved