A062801 -id:A062801 - cp's OEIS frontend

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

John W. Layman, Jan 25 2001

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

Cf. A062801, A134506.

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 *)

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

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

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

A171503 Number of 2 X 2 integer matrices with entries from {0,1,...,n} having determinant 1.

Original entry on oeis.org

0, 3, 7, 15, 23, 39, 47, 71, 87, 111, 127, 167, 183, 231, 255, 287, 319, 383, 407, 479, 511, 559, 599, 687, 719, 799, 847, 919, 967, 1079, 1111, 1231, 1295, 1375, 1439, 1535, 1583, 1727, 1799, 1895, 1959, 2119, 2167, 2335, 2415, 2511, 2599

Offset: 0

Jacob A. Siehler, Dec 10 2009

nonn

Number of distinct solutions to k*x+h=0, where |h|<=n and k=1,2,...,n. - Giovanni Resta, Jan 08 2013.

Number of reduced rational numbers r/s with |r|<=n and 0Juan M. Marquez, Apr 13 2015

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A062801, A000010, A018805. Differences are A002246.

See A326354 for an essentially identical sequence.

with(numtheory):
a:= proc(n) option remember;
       `if`(n<2, [0, 3][n+1], a(n-1) + 4*phi(n))
    end:
seq(a(n), n=0..60);

a[n_]:=Count[Det/@(Partition[ #,2]&/@Tuples[Range[0,n],4]),1]
(* Second program: *)
a[0] = 0; a[1] = 3; a[n_] := a[n] = a[n-1] + 4*EulerPhi[n];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Jun 16 2018 *)

a(n)=(n>0)+2*sum(k=1, n, moebius(k)*(n\k)^2) \\ Charles R Greathouse IV, Apr 20 2015

from functools import lru_cache
@lru_cache(maxsize=None)
def A171503(n): # based on second formula in A018805
    if n == 0:
        return 0
    c, j = 0, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*(A171503(k1)-1)//2
        j, k1 = j2, n//j2
    return 2*(n*(n-1)-c+j) - 1 # Chai Wah Wu, Mar 25 2021

Recursion: a(n) = a(n - 1) + 4*phi(n) for n > 1, with phi being Euler's totient function. - Juan M. Marquez, Jan 19 2010

a(n) = 4 * A002088(n) - 1 for n >= 1. - Robert Israel, Jun 01 2014

Edited by Alois P. Heinz, Jan 19 2011

A187462 T(n,k)=Number of (n+1)X(n+1) 0..k arrays with all the 2X2 subblocks nonsingular and the array of 2X2 subblock determinants symmetric.

Original entry on oeis.org

6, 50, 26, 192, 1236, 144, 512, 13726, 89618, 1272, 1126, 84048, 5098224, 11684688, 15192, 2146, 325932, 100936174, 3387715318, 6112320620, 286392, 3760, 1118854, 1061948616, 216668424092, 19708386669940, 4871877392202, 7648032, 6112

Offset: 1

R. H. Hardin Mar 10 2011

Table starts
.........6................50................192..............512..........1126
........26..............1236..............13726............84048........325932
.......144.............89618............5098224........100936174....1061948616
......1272..........11684688.........3387715318.....216668424092.5212823523842
.....15192........6112320620.....19708386669940.7540469990506332
....286392.....4871877392202.150657364854061134
...7648032.17264863350746654
.314011868

			Some k=3 solutions for 4X4
..0..2..2..0....0..2..3..3....0..2..2..3....0..2..2..2....0..2..2..0
..3..2..0..2....1..2..0..1....3..2..0..1....2..2..0..2....2..3..0..3
..2..0..1..2....3..0..2..3....2..0..3..3....2..0..1..1....2..0..1..0
..0..2..2..3....2..1..3..0....1..1..2..0....3..2..2..0....2..3..3..3

R. H. Hardin, Table of n, a(n) for n = 1..40

Row 1 is A062801

A020478 Number of singular 2 X 2 matrices over Z(n) (i.e., with determinant = 0).

Original entry on oeis.org

1, 10, 33, 88, 145, 330, 385, 736, 945, 1450, 1441, 2904, 2353, 3850, 4785, 6016, 5185, 9450, 7201, 12760, 12705, 14410, 12673, 24288, 18625, 23530, 26001, 33880, 25201, 47850, 30721, 48640, 47553, 51850, 55825, 83160, 51985, 72010, 77649, 106720

Offset: 1

David W. Wilson

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Cf. A005353, A059306, A062801, A069097, A102631, A240547.

f[p_, e_] := p^(2*e - 1)*(p^(e + 1) + p^e - 1); a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 22 2020 *)

a(n)=if(n<1, 0, direuler(p=2, n, (1-p*X)/((1-p^2*X)*(1-p^3*X)))[n])

a(n)=local(c=0); forvec(x=vector(4,k,[1,n]),c+=((x[1]*x[2]-x[3]*x[4])%n==0)); c

From Vladeta Jovovic, Apr 22 2002: (Start)
a(n) = n^4 - A005353(n).
Multiplicative with a(p^e) = p^(2*e - 1)*(p^(e+1) + p^e - 1). (End)
Dirichlet g.f.: zeta(s-2)*zeta(s-3)/zeta(s-1).
A102631(n) | a(n). - R. J. Mathar, Mar 30 2011
Sum_{k=1..n} a(k) ~ Pi^2 * n^4 / (24*Zeta(3)). - Vaclav Kotesovec, Jan 31 2019
From Piotr Rysinski, Sep 11 2020: (Start)
a(n) = n * A069097(n).
Proof: a(n) is multiplicative with a(p^e) = p^(2*e - 1)*(p^(e+1) + p^e - 1), A069097(n) is multiplicative with A069097(p^e) = p^(e-1)*(p^e*(p+1)-1), so a(p^e) = p^e*A069097(p^e). (End)

A187528 T(n,k)=Number of (n+1)X(n+1) 0..k arrays with each 2X2 subblock determinant nonzero and the array of 2X2 subblock determinants symmetric under 90 degree rotation.

Original entry on oeis.org

6, 50, 0, 192, 8, 104, 512, 74, 3790, 0, 1126, 280, 35026, 1476, 9346, 2146, 784, 220802, 20554, 3607148, 0, 3760, 1970, 727842, 273102, 164460556, 296232, 4278774, 6112, 3580, 2327574, 992410

Offset: 1

R. H. Hardin Mar 10 2011

Table starts
.......6......50.......192....512...1126....2146.3760.6112
.......0.......8........74....280....784....1970.3580
.....104....3790.....35026.220802.727842.2327574
.......0....1476.....20554.273102.992410
....9346.3607148.164460556
.......0..296232
.4278774

			Some k=3 solutions for 5X5
..1..2..3..1..1....1..2..3..1..3....3..2..1..1..2....1..1..3..2..3
..0..2..2..0..2....0..3..3..0..3....1..0..1..2..2....3..0..3..3..3
..1..2..3..1..2....1..2..3..1..3....2..1..1..1..2....2..1..2..1..2
..1..0..1..1..0....2..1..3..2..3....2..2..1..0..1....3..3..3..0..3
..2..2..3..1..2....1..2..3..1..3....2..1..1..2..1....2..1..2..1..2

Row 1 is A062801

A005353 Number of 2 X 2 matrices with entries mod n and nonzero determinant.

Original entry on oeis.org

0, 6, 48, 168, 480, 966, 2016, 3360, 5616, 8550, 13200, 17832, 26208, 34566, 45840, 59520, 78336, 95526, 123120, 147240, 181776, 219846, 267168, 307488, 372000, 433446, 505440, 580776, 682080, 762150, 892800, 999936, 1138368, 1284486

Offset: 1

N. J. A. Sloane, R. K. Guy

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

T. D. Noe, Table of n, a(n) for n=1..1000
Richard K. Guy, Letter to N. J. A. Sloane, Sep 1989.

Cf. A020478, A059306, A062801.

Table[cnt=0; Do[m={{a, b}, {c, d}}; If[Det[m, Modulus->p] > 0, cnt++ ], {a, 0, p-1}, {b, 0, p-1}, {c, 0, p-1}, {d, 0, p-1}]; cnt, {p, 37}] (* T. D. Noe, Jan 12 2006 *)
f[p_, e_] := p^(2*e - 1)*(p^(e + 1) + p^e - 1); a[1] = 0; a[n_] := n^4 - Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 31 2023 *)

a(n) = {my(f = factor(n), p, e); n^4 - prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2];  p^(2*e - 1)*(p^(e + 1) + p^e - 1));} \\ Amiram Eldar, Oct 31 2023

a(n) = n^4 - A020478(n).

For prime n, a(n) = (n^2-1)(n-1)n. - T. D. Noe, Jan 12 2006

More terms from T. D. Noe, Jan 12 2006

A187670 T(n,k)=Number of (n+1)X(n+1) 0..k arrays with each 2X2 subblock nonsingular and the array of 2X2 subblock determinants symmetric about the diagonal and antidiagonal.

Original entry on oeis.org

6, 50, 24, 192, 550, 112, 512, 4044, 11282, 974, 1126, 17716, 174292, 612204, 9928, 2146, 56080, 1549398, 36461174, 65533922, 197152, 3760, 150166, 6839146, 835707780

Offset: 1

R. H. Hardin Mar 12 2011

Table starts
......6.......50......192.......512....1126...2146.3760
.....24......550.....4044.....17716...56080.150166
....112....11282...174292...1549398.6839146
....974...612204.36461174.835707780
...9928.65533922
.197152

			Some k=3 solutions for 5X5
..0..2..2..0..3....0..2..2..2..3....0..2..2..2..1....0..2..2..2..3
..2..1..0..1..0....2..2..0..1..1....2..2..0..1..1....2..2..0..1..1
..2..0..3..0..2....2..0..2..0..2....2..0..1..1..3....2..0..1..0..2
..0..1..0..1..2....0..1..0..2..2....3..1..0..2..2....0..1..0..2..3
..3..2..2..2..0....1..1..2..3..1....2..1..2..2..0....1..1..2..2..1

Row 1 is A062801

A187699 T(n,k)=Number of (n+1)X(n+1) 0..k arrays with each 2X2 subblock determinant nonzero and the array of 2X2 subblock determinants symmetric under horizontal and vertical reflection.

Original entry on oeis.org

6, 50, 0, 192, 8, 104, 512, 74, 7954, 0, 1126, 280, 87258, 1548, 9350, 2146, 784, 692410, 22738, 15128848, 0, 3760, 1970, 2363474, 320674, 1089964504, 454640, 4280018, 6112, 3580, 10839282, 1147854

Offset: 1

R. H. Hardin Mar 12 2011

Table starts
.......6.......50........192....512....1126.....2146.3760.6112
.......0........8.........74....280.....784.....1970.3580
.....104.....7954......87258.692410.2363474.10839282
.......0.....1548......22738.320674.1147854
....9350.15128848.1089964504
.......0...454640
.4280018

			Some k=3 solutions for 5X5
..1..0..1..1..2....3..2..3..1..0....2..1..3..2..3....1..2..3..1..2
..0..1..1..0..1....3..3..3..0..3....1..2..3..1..3....2..1..3..2..1
..1..1..2..1..2....2..1..2..1..1....2..1..3..2..3....1..2..3..1..2
..1..0..1..1..1....3..0..3..3..0....3..0..3..3..3....0..3..3..0..3
..3..1..2..1..2....3..1..3..2..1....3..1..2..1..2....1..2..3..1..2

Row 1 is A062801

Row 2 is A187529

A064276 Number of 2 X 2 singular integer matrices with elements from {0,...,n} up to row and column permutation.

Original entry on oeis.org

1, 5, 13, 25, 42, 62, 90, 118, 155, 195, 243, 287, 352, 404, 472, 548, 629, 697, 797, 873, 986, 1094, 1202, 1294, 1443, 1559, 1687, 1823, 1984, 2100, 2296, 2420, 2597, 2769, 2937, 3125, 3366, 3514, 3702, 3906, 4167, 4331, 4611, 4783, 5040, 5320, 5548

Offset: 0

Vladeta Jovovic, Sep 24 2001

nonn

			There are 5 binary singular matrices up to row and column permutation:
[0 0] [1 0] [1 1] [1 0] [1 1]
[0 0] [0 0] [0 0] [1 0] [1 1].

Cf. A059306, A062801, A059976.

a(n) = (A059306(n)+(n+1)*(2*n+3))/4.

More terms from David Wasserman, Jul 16 2002

A064368 Number of 2 X 2 symmetric singular matrices with entries from {0,...,n}.

Original entry on oeis.org

1, 4, 7, 10, 15, 18, 21, 24, 29, 36, 39, 42, 47, 50, 53, 56, 65, 68, 75, 78, 83, 86, 89, 92, 97, 108, 111, 118, 123, 126, 129, 132, 141, 144, 147, 150, 163, 166, 169, 172, 177, 180, 183, 186, 191, 198, 201, 204, 213, 228, 239, 242, 247, 250, 257, 260, 265, 268

Offset: 0

Vladeta Jovovic, Sep 27 2001

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000010, A000188, A064276, A059306, A062801, A059976, A039623.

Cf. A001620, A306016.

f[p_, e_] := p^Floor[e/2]; a[0] = 0; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; With[{max = 100}, 1 + Range[0, max] + 2 * Accumulate[Array[a, max + 1, 0]]] (* Amiram Eldar, Nov 07 2024 *)

a(n) = n + 1 + 2*sum(k=1, n, sumdiv(k, d, issquare(d)*eulerphi(sqrtint(d)))) \\ Michel Marcus, Jun 17 2013

a(n) = n + 1 + 2*Sum_{k=1..n} Sum_{d^2|k} phi(d), where phi = Euler totient function A000010.

a(n) ~ (n/zeta(2)) * (log(n) + 3*gamma - 1 + zeta(2) - 2*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 07 2024

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Python

Formula

Extensions

A171503 Number of 2 X 2 integer matrices with entries from {0,1,...,n} having determinant 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

Extensions

A187462 T(n,k)=Number of (n+1)X(n+1) 0..k arrays with all the 2X2 subblocks nonsingular and the array of 2X2 subblock determinants symmetric.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A020478 Number of singular 2 X 2 matrices over Z(n) (i.e., with determinant = 0).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A187528 T(n,k)=Number of (n+1)X(n+1) 0..k arrays with each 2X2 subblock determinant nonzero and the array of 2X2 subblock determinants symmetric under 90 degree rotation.

Views

Author

Keywords

Comments

Examples

Crossrefs

A005353 Number of 2 X 2 matrices with entries mod n and nonzero determinant.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A187670 T(n,k)=Number of (n+1)X(n+1) 0..k arrays with each 2X2 subblock nonsingular and the array of 2X2 subblock determinants symmetric about the diagonal and antidiagonal.

Views

Author

Keywords

Comments

Examples

Crossrefs

A187699 T(n,k)=Number of (n+1)X(n+1) 0..k arrays with each 2X2 subblock determinant nonzero and the array of 2X2 subblock determinants symmetric under horizontal and vertical reflection.

Views

Author

Keywords

Comments

Examples

Crossrefs

A064276 Number of 2 X 2 singular integer matrices with elements from {0,...,n} up to row and column permutation.

Views