A020478 -id:A020478 - cp's OEIS frontend

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

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

A240547 Number of non-congruent solutions of x^2 + y^2 + z^2 + t^2 == 0 mod n.

Original entry on oeis.org

1, 8, 33, 32, 145, 264, 385, 128, 945, 1160, 1441, 1056, 2353, 3080, 4785, 512, 5185, 7560, 7201, 4640, 12705, 11528, 12673, 4224, 18625, 18824, 26001, 12320, 25201, 38280, 30721, 2048, 47553, 41480, 55825, 30240, 51985, 57608, 77649, 18560, 70561, 101640

Offset: 1

Laszlo Toth, Apr 07 2014

			For n=2 the a(2)=8 solutions are (0,0,0,0), (1,1,0,0), (1,0,1,0), (1,0,0,1), (0,1,1,0), (0,1,0,1), (0,0,1,1), (1,1,1,1).

David A. Corneth, Table of n, a(n) for n = 1..10000
László Tóth, Counting solutions of quadratic congruences in several variables revisited, arXiv preprint arXiv:1404.4214 [math.NT], 2014.
László Tóth, Counting Solutions of Quadratic Congruences in Several Variables Revisited, J. Int. Seq. 17 (2014), Article 14.11.6.
Index to sequences related to sums of squares.

Cf. A020478, A069097, A086933, A087687, A208895, A229179.

A240547 := proc(n) local a, x, y, z, t ; a := 0 ; for x from 0 to n-1 do for y
from 0 to n-1 do for z from 0 to n-1 do for t from 0 to n-1 do if
(x^2+y^2+z^2+t^2) mod n = 0 mod n then a := a+1 ; fi; od; od ; od; od;
a ; end proc;
# alternative
A240547 := proc(n)
    a := 1;
    for pe in ifactors(n)[2] do
        p := op(1,pe) ;
        e := op(2,pe) ;
        if p = 2 then
            a := a*p^(2*e+1) ;
        else
            a := a* p^(2*e-1)*(p^(e+1)+p^e-1) ;
        end if;
    end do:
    a ;
end proc:
seq(A240547(n),n=1..100) ; # R. J. Mathar, Jun 25 2018

b[2, e_] := 2^(2 e + 1);
b[p_, e_] := p^(2 e - 1)*(p^(e + 1) + p^e - 1);
a[n_] := Times @@ b @@@ FactorInteger[n];
Array[a, 42] (* Jean-François Alcover, Dec 05 2017 *)

a(n) = my(m); if( n<1, 0, forvec( v = vector(4, i, [0, n-1]), m += (0 == norml2(v)%n))); m /* Michael Somos, Apr 07 2014 */

a(n) = {my(f = factor(n), res = 1, start = 1, p, e, i); if(n % 2 == 0, res = 1<<(f[1,2]<<1+1); start = 2); for(i = start, #f~, p = f[i, 1]; e = f[i, 2]; res*=(p^(e<<1-1)*(p^(e+1)+p^e-1))); res} \\ David A. Corneth, Jul 22 2018

Multiplicative, with a(2^e) = 2^(2e+1) for e>=1, a(p^e) = p^(2e-1)*(p^(e+1)+p^e-1) for p > 2, e>=1.
For odd n, a(n) = A069097(n)*n = A020478(n). - R. J. Mathar, Jun 23 2018
Sum_{k=1..n} a(k) ~ c * n^4 + O(n^3 * log(n)), where c = 5*Pi^2/(168*zeta(3)) = 0.244362... (Tóth, 2014). - Amiram Eldar, Oct 18 2022

A115075 Number of 2 X 2 symmetric matrices over Z(n) having determinant 0.

Original entry on oeis.org

1, 4, 9, 20, 25, 36, 49, 80, 99, 100, 121, 180, 169, 196, 225, 352, 289, 396, 361, 500, 441, 484, 529, 720, 725, 676, 891, 980, 841, 900, 961, 1408, 1089, 1156, 1225, 1980, 1369, 1444, 1521, 2000, 1681, 1764, 1849, 2420, 2475, 2116, 2209, 3168, 2695, 2900

Offset: 1

T. D. Noe, Jan 12 2006

Cf. A020478 (number of singular 2 X 2 matrices over Z(n)), A115077.

Cf. A002117, A013662.

Table[cnt=0; Do[m={{a, b}, {b, c}}; If[Det[m, Modulus->n]==0, cnt++ ], {a, 0, n-1}, {b, 0, n-1}, {c, 0, n-1}]; cnt, {n, 50}]
f[p_, e_] := p^e*(p^e + p^(e-1) - p^(Ceiling[e/2] - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]  (* Amiram Eldar, Oct 31 2023 *)

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

a(n) = n^3 - A115077(n).
For squarefree n, a(n) = n^2.
Multiplicative with a(p^e) = p^(e)*(p^(e)+p^(e-1)-p^(ceiling(e/2)-1)).
From Amiram Eldar, Oct 31 2023: (Start)
Dirichlet g.f.: zeta(s-2) * zeta(2*s-3) / zeta(2*s-2).
Sum_{k=1..n} a(k) ~ (zeta(3)/(3*zeta(4))) * n^3. (End)

A181107 Triangle read by rows: T(n,k) is the number of 2 X 2 matrices over Z(n) having determinant congruent to k mod n, 1 <= n, 0 <= k <= n-1.

Original entry on oeis.org

1, 10, 6, 33, 24, 24, 88, 48, 72, 48, 145, 120, 120, 120, 120, 330, 144, 240, 198, 240, 144, 385, 336, 336, 336, 336, 336, 336, 736, 384, 576, 384, 672, 384, 576, 384, 945, 648, 648, 864, 648, 648, 864, 648, 648, 1450, 720, 1200, 720, 1200, 870, 1200, 720, 1200, 720

Offset: 1

Erdos Pal, Oct 03 2010

The n-th row is {T(n,0),T(n,1),...,T(n,n-1)}.
Let m denote the prime power p^e.
T(m,0) = A020478(m) = (p^(e+1) + p^e-1)*p^(2*e-1).
T(m,1) = A000056(m) = (p^2-1)*p^(3*e-2).
T(prime(n),1) = A127917(n).
Sum_{k=1..n-1} T(n,k) = A005353(n).
T(n,1) = n*A007434(n) for n>=1 because A000056(n) = n*Jordan_Function_J_2(n).
T(2^n,1) = A083233(n) = A164640(2n) for n>=1. Proof: a(n):=T(2^n,1); a(1)=6, a(n)=8*a(n-1); A083233(1)=6 and A083233(n) is a geometric series with ratio 8 (because of its g.f.), too; A164640 = {b(1)=1, b(2)=6, b(n)=8*b(n-2)}.
T(2^n,0) = A165148(n) for n>=0, because  2*T(2^n,0) = (3*2^n-1)*4^n.
T(2^e,2) = A003951(e) for 2 <= e. Proof: T(2^e,2) = 9*8^(e-1) is a series with ratio 8 and initial term 72, as A003951(2...inf) is.
Working with consecutive powers of a prime p, we need a definition (0 <= i < e):
N(p^e,i):=#{k: 0 < k < p^e, gcd(k,p^e) = p^i} = (p-1)*p^(e-1-i). We say that these k's belong to i (respect to p^e). Note that N(p^e,0) = EulerPhi(p^e), and if 0 < k < p^e then gcd(k,p^e) = gcd(k,p^(e+1)). Let T(p^e,[i]) denote the common value of T(p^e,k)'s, where k's belong to i (q.v.PROGRAM); for example, T(p^e,[0]) = T(p^e,1). The number of the 2 X 2 matrices over Z(p^e), T(p^e,0) + Sum_{i=0..e-1} T(p^e,[i])*N(p^e,i) = p^(4e) will be useful.
On the hexagon property: Let prime p be given and let T(p^e,[0]), T(p^e,[1]), T(p^e,[2]), ..., T(p^e,[e-2]), T(p^e,[e-1]) form the e-th row of a Pascal-like triangle, e>=1. Let denote X(r,s) an element of the triangle and its value T(p^r,[s]). Let positive integers a and b given, so that the entries A(m-a,n-b), B(m-a,n), C(m,n+a), D(m+b,n+a), E(m+b,n), F(m,n-b) of the triangle form a hexagon spaced around T(p^m,[n]); if a=b=1 then they surround it. If A*C*E = B*D*F, then we say that the triangle T(.,.) has the "hexagon property". (In the case of binomial coefficients X(r,s) = COMB(r,s), the "hexagon property" holds (see [Gupta]) and moreover gcd(A,C,E) = gcd(B,D,F) (see [Hitotumatu & Sato]).)
Corollary 2.2 in [Brent & McKay] says that, for the d X d matrices over Z(p^e), (mutatis mutandis) T_d(p^e,0) = K*(1-P(d+e-1)/P(e-1)) and T_d(p^e,[i]) = K*(q^e)*((1-q^d)/(1-q))*P(d+i-1)/P(i), where q=1/p, K=(p^e)^(d^2), P(t) = Product_{j=1..t} (1-q^j), P(0):=1. (For the case d=2, we have T(p^e,[i]) = (p+1)*(p^(i+1)-1)*p^(3*e-i-2).) Due to [Brent & McKay], it can be simply proved that for d X d matrices the "hexagon property" is true. The formulation implies an obvious generalization: For the entries A(r,u), B(r,v), C(s,w), D(t,w), E(t,v), F(s,u) of the T_d(.,.)-triangle, a hexagon-like property A*C*E = B*D*F holds. This is false in general for the COMB(.,.)-triangle.
Another (rotated-hexagon-like) property: for the entries A(m-b1,n), B(m-a1,n+c2), C(m+a2,n+c2), D(m+b2,n), E(m+a2,n-c1), F(m-a1,n-c1) of the T_d(.,.)-triangle, the property A*C*E = B*D*F holds, if and only if 2*(a1 + a2) = b1 + b2. This is also in general false for COMB(.,.)-triangle.

			From _Andrew Howroyd_, Jul 16 2018: (Start)
Triangle begins:
    1;
   10,   6;
   33,  24,  24;
   88,  48,  72,  48;
  145, 120, 120, 120, 120;
  330, 144, 240, 198, 240, 144;
  385, 336, 336, 336, 336, 336, 336;
  736, 384, 576, 384, 672, 384, 576, 384;
  945, 648, 648, 864, 648, 648, 864, 648, 648;
  ... (End)

Erdos Pal, Rows n=1..100 of triangle, flattened
Richard P. Brent and Brendan D. McKay, Determinants and ranks of random matrices over Z_m, Discrete Mathematics 66 (1987) pp. 35-49.
A. K. Gupta, Generalized hidden hexagon squares, The Fibonacci Quarterly, Vol 12, Number 1, Feb.1974, pp. 45-46.
S. Hitotumatu, D. Sato, Star of David theorem (I), The Fibonacci Quarterly, Vol 13, Number 1, Feb.1975, p. 70.

Column k=0 is A020478.
Column k=1 is A000056.
Row sums are A005353.
Cf. A000252, A127917.

  (* computing T(p^e,k) ; p=prime, 1<=e, 0<=k

S(p,e)={my(u=vector(p^e)); my(t=(p-1)*p^(e-1)); u[1] = p^e + e*t; for(j=1, p^e-1, u[j+1] = t*(1+valuation(j, p))); vector(#u, k, sum(j=0, #u-1, u[j + 1]*u[(j+k-1) % #u + 1]))}
T(n)={my(f=factor(n), v=vector(n,i,1)); for(i=1, #f~, my(r=S(f[i,1], f[i,2])); for(j=0, #v-1, v[j + 1] *= r[j % #r + 1])); v}
for(n=1, 10, print(T(n))); \\ Andrew Howroyd, Jul 16 2018

T(a*b,k) = T(a,(k mod a))*T(b,(k mod b)) if gcd(a,b) = 1.

Sum_{k=1..n-1, gcd(k,n)=1} T(n,k) = A000252(n). - Andrew Howroyd, Jul 16 2018

Terms a(24)-a(55) from b-file by Andrew Howroyd, Jul 16 2018

A345754 Number of 2 X 2 matrices over Z_n whose permanent equals their determinant.

Original entry on oeis.org

1, 16, 45, 192, 225, 720, 637, 2048, 1701, 3600, 2541, 8640, 4225, 10192, 10125, 20480, 9537, 27216, 13357, 43200, 28665, 40656, 23805, 92160, 40625, 67600, 59049, 122304, 47937, 162000, 58621, 196608, 114345, 152592, 143325, 326592, 99937, 213712, 190125

Offset: 1

José María Grau Ribas, Jul 19 2021

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A020478, A344372.

Cf. A018804.

a[n_] := a[n] =  Sum[If[Mod[Permanent[{{a, b}, {c, d}}] - Det[{{a, b}, {c, d}}],n] == 0, 1, 0], {a, 0, n - 1}, {b, 0, n - 1}, {c, 0, n - 1}, {d,0, n - 1}] ; Array[a,22]
f[p_, e_] := p^(3*e - 1)*((p - 1)*e + p); f[2, e_] := (e + 1)*8^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Dec 06 2022 *)

a(n) =  A344372(n) * n^2 (conjectured).
From Sebastian Karlsson, Aug 31 2021: (Start)
The formula is correct. Proof:
Let A = ([a, b], [c, d]) be an arbitrary 2 X 2 matrix over Z_n. So det(A) = a*d - b*c and perm(A) = a*d + b*c. Then, det(A) = perm(A) iff -b*c = b*c, i.e., 2*b*c = 0.
Suppose first that n is odd. Then 2*b*c = 0 iff b*c = 0. The number of solutions to this equation over Z_n is A018804(n). Furthermore, the value of a and b in A can be anything, so there are n possible choices for a and n possible choices for b. Thus, there are n*n*A018804(n) = n^2 * A344372(n) matrices A over Z_n such that det(A) = perm(A).
Suppose now that n is even. Then 2*b*c = 0 in Z_n iff b'*c' = 0 in Z_{n/2}, where b' and c' are b and c reduced modulo n/2. The latter equation has A018804(n/2) distinct solutions in Z_{n/2}. As the preimage of both b' and c' contains precisely 2 elements each, the number of solutions to 2*b*c = 0 in Z_n is 2*2*A018804(n/2). Hence, a(n) = n*n*4*A018804(n/2) = n^2 * A344372(n). Q.E.D.
The formula implies that the sequence is multiplicative with a(2^e) = (e+1)*8^e, a(p^e) = p^(3*e-1)*((p-1)*e+p) for odd primes p. (End)

cp's OEIS Frontend

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

A240547 Number of non-congruent solutions of x^2 + y^2 + z^2 + t^2 == 0 mod n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A115075 Number of 2 X 2 symmetric matrices over Z(n) having determinant 0.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A181107 Triangle read by rows: T(n,k) is the number of 2 X 2 matrices over Z(n) having determinant congruent to k mod n, 1 <= n, 0 <= k <= n-1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Other

PARI

Formula

Extensions

A345754 Number of 2 X 2 matrices over Z_n whose permanent equals their determinant.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula