A059306 Number of 2 X 2 singular integer matrices with elements from {0,...,n}.

1, 10, 31, 64, 113, 170, 255, 336, 449, 570, 719, 848, 1057, 1210, 1423, 1664, 1921, 2122, 2447, 2672, 3041, 3386, 3727, 4000, 4497, 4858, 5263, 5696, 6225, 6570, 7231, 7600, 8177, 8730, 9263, 9872, 10689, 11130, 11727, 12384, 13265, 13754, 14703

Offset: 0

Author

John W. Layman, Jan 25 2001

Keywords

• nonn
• nice

Links

• Vincenzo Librandi and Chai Wah Wu, Table of n, a(n) for n = 0..10000 (terms for n = 0..100 from Vincenzo Librandi)

Crossrefs

Cf. A062801, A134506.

Programs

Mathematica

a[0] = 1; a[n_] := Table[{w, x, y, z} /. {ToRules[ Reduce[0 <= x <= n && 0 <= y <= n && 0 <= z <= n && w*z - x*y == 0, {x, y, z}, Integers]] }, {w, 0, n}] // Flatten[#, 1]& // Length; Table[Print[an = a[n]]; an, {n, 0, 42}] (* Jean-François Alcover, Oct 11 2013 *)

Python

from math import gcd
def A059306(n): return (2*n+1)*(n+1) + 4*sum(gcd(i, j) for i in range(1, n+1) for j in range(i, n+1)) # David Radcliffe, Aug 13 2025

Formula

a(n) = A134506(n) + (2n+1)^2. Shi's result (see formula section in A134506) shows that a(n) = kn^2 log n + cn^2 + O(n^e) where k = 12/Pi^2, e > 547/416 = 1.3149..., and c = 4.5113... - Chai Wah Wu, Nov 28 2016

a(n) = 4*A272718(n) + 2*n^2 + 3*n + 1. - David Radcliffe, Aug 13 2025

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Jan 09 2003