A208895 -id:A208895 - cp's OEIS frontend

A087784 Number of solutions to x^2 + y^2 + z^2 = 1 mod n.

1, 4, 6, 24, 30, 24, 42, 96, 54, 120, 110, 144, 182, 168, 180, 384, 306, 216, 342, 720, 252, 440, 506, 576, 750, 728, 486, 1008, 870, 720, 930, 1536, 660, 1224, 1260, 1296, 1406, 1368, 1092, 2880, 1722, 1008, 1806, 2640, 1620, 2024, 2162, 2304, 2058, 3000

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 06 2003

Michael De Vlieger, 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, Journal of Integer Sequences, 17 (2014), Article 14.11.6.

Cf. A006752, A060968, A087687, A208895, A264083.

Table[With[{f = FactorInteger[n][[All, 1]]}, Apply[Times, Map[1 + 1/# &, Select[f, Mod[#, 4] == 1 &]]] Apply[Times, Map[1 - 1/# &, Select[f, Mod[#, 4] == 3 &]]] (1 + Boole[Divisible[n, 4]]/2) n^2] - Boole[n == 1], {n, 50}] (* Michael De Vlieger, Feb 15 2018 *)

a(n) = {my(f=factor(n)); if ((n % 4), 1, 3/2)*n^2*prod(k=1, #f~, p = f[k,1]; m = p % 4; if (m==1, 1+1/p, if (m==3, 1-1/p, 1)));} \\ Michel Marcus, Feb 14 2018

a(n) = n^2 * (3/2 if 4|n) * Product_{primes p == 1 mod 4 dividing n} (1+1/p) * Product_{primes p == 3 mod 4 dividing n} (1-1/p). - Bjorn Poonen, Dec 09 2003

Sum_{k=1..n} a(k) ~ c * n^3 + O(n^2 * log(n)), where c = 36*G/Pi^4 = 0.338518..., where G is Catalan's constant (A006752) (Tóth, 2014). - Amiram Eldar, Oct 18 2022

More terms from David Wasserman, Jun 17 2005

A071302 a(n) = (1/2) * (number of n X n 0..2 matrices M with MM' mod 3 = I, where M' is the transpose of M and I is the n X n identity matrix).

Original entry on oeis.org

1, 4, 24, 576, 51840, 13063680, 9170703360, 19808719257600, 131569513308979200, 2600339861038664908800, 152915585868239728626892800, 27051378802435080953011843891200, 14395932257291877030764312963579904000

Offset: 1

R. H. Hardin, Jun 11 2002

nonn

Also, number of n X n orthogonal matrices over GF(3) with determinant 1. - Max Alekseyev, Nov 06 2022

			From _Petros Hadjicostas_, Dec 17 2019: (Start)
For n = 2, the 2*a(2) = 8 n X n matrices M with elements in {0, 1, 2} that satisfy MM' mod 3 = I are the following:
(a) With 1 = det(M) mod 3:
[[1,0],[0,1]];  [[0,1],[2,0]]; [[0,2],[1,0]]; [[2,0],[0,2]].
This is the abelian group SO(2, Z_3). See the comments for sequence A060968.
(b) With 2 = det(M) mod 3:
[[0,1],[1,0]];  [[0,2],[2,0]]; [[1,0],[0,2]]; [[2,0],[0,1]].
Note that, for n = 3, we have 2*a(3) = 2*24 = 48 = A264083(3). (End)

Jianing Song, Structure of the group SO(2,Z_n).
László Tóth, Counting solutions of quadratic congruences in several variables revisited, arXiv:1404.4214 [math.NT], 2014.
László Tóth, Counting Solutions of Quadratic Congruences in Several Variables Revisited, J. Int. Seq. 17 (2014), #14.11.6.
Jessie MacWilliams, Orthogonal Matrices Over Finite Fields, The American Mathematical Monthly 76:2 (1969), 152-164.

Cf. A003053, A003920, A060968, A071303, A071304, A071305, A071306, A071307, A071308, A071309, A071310, A071900, A087784, A208895, A264083, A318609.

FoldList[Times, 1, LinearRecurrence[{3, -3, 9}, {4, 6, 24}, 12]] (* Amiram Eldar, Jun 22 2025 *)

{ a071302(n) = my(t=n\2); prod(i=0,t-1,3^(2*t)-3^(2*i)) * if(n%2,3^t,1/(3^t+(-1)^t)); } \\ Max Alekseyev, Nov 06 2022

a(2k+1) = 3^k * Product_{i=0..k-1} (3^(2k) - 3^(2i)); a(2k) = (3^k + (-1)^(k+1)) * Product_{i=1..k-1} (3^(2k) - 3^(2i)) (see MacWilliams, 1969). - Max Alekseyev, Nov 06 2022

a(n+1) = a(n) * A318609(n+1) for n >= 1. - conjectured by Petros Hadjicostas, Dec 18 2019; proved based on the explicit formula by Max Alekseyev, Nov 06 2022

Terms a(8) onward from Max Alekseyev, Nov 06 2022

A071303 1/2 times the number of n X n 0..3 matrices M with MM' mod 4 = I, where M' is the transpose of M and I is the n X n identity matrix.

Original entry on oeis.org

1, 8, 192, 12288, 1966080, 1509949440, 5411658792960

Offset: 1

R. H. Hardin, Jun 11 2002

It seems that a(n) = n! * 2^(binomial(n+1,2) - 1) for n = 1, 2, 3, 4, 5, while for n = 6, a(n) is twice this number. The number n! * 2^(binomial(n+1,2) - 1) appears in Proposition 6.1 in Eriksson and Linusson (2000) as an upper bound to the number of three-dimensional permutation arrays of size n (see column k = 3 of A330490). - Petros Hadjicostas, Dec 16 2019

a(7) = 7! * 2^30. - Sean A. Irvine, Jul 11 2024

			From _Petros Hadjicostas_, Dec 16 2019: (Start)
For n = 2, here are the 2*a(2) = 16 2 x 2 matrices M with elements in {0,1,2,3} that satisfy MM'  mod 4 = I:
(a) With 1 = det(M) mod 4:
  [[1,0],[0,1]]; [[0,1],[3,0]]; [[0,3],[1,0]]; [[1,2],[2,1]];
  [[2,1],[3,2]]; [[2,3],[1,2]]; [[3,0],[0,3]]; [[3,2],[2,3]].
These form the abelian group SO(2, Z_n). See the comments for sequence A060968.
(b) With 3 = det(M) mod 4:
  [[0,1],[1,0]]; [[0,3],[3,0]]; [[1,0],[0,3]];  [[1,2],[2,3]];
  [[2,1],[1,2]]; [[2,3],[3,2]]; [[3,0],[0,1]];  [[3,2],[2,1]].
Note that, for n = 3, we have 2*a(3) = 2*192 = 384 = A264083(4). (End)

Kimmo Eriksson and Svante Linusson, A combinatorial theory of higher-dimensional permutation arrays, Adv. Appl. Math. 25(2) (2000a), 194-211; see Proposition 6.1 (p. 210).
Sean A. Irvinne, Java program (github)
Jianing Song, Structure of the group SO(2,Z_n).
László Tóth, Counting solutions of quadratic congruences in several variables revisited, arXiv:1404.4214 [math.NT], 2014.
László Tóth, Counting Solutions of Quadratic Congruences in Several Variables Revisited, J. Int. Seq. 17 (2014), #14.11.6.

Cf. A003920, A060968, A071302, A071304, A071305, A071306, A071307, A071308, A071309, A071310, A071900, A087784, A208895, A229136, A264083, A330490.

a(7) from Sean A. Irvine, Jul 11 2024

A071304 a(n) = (1/2) * (number of n X n 0..4 matrices M with MM' mod 5 = I, where M' is the transpose of M and I is the n X n identity matrix).

Original entry on oeis.org

1, 4, 120, 14400, 9360000, 29016000000, 457002000000000, 35646156000000000000, 13946558535000000000000000, 27230655539587500000000000000000, 266009466302345390625000000000000000000, 12987912192212013697265625000000000000000000000

Offset: 1

R. H. Hardin, Jun 11 2002

nonn

Also, number of n X n orthogonal matrices over GF(5) with determinant 1. - Max Alekseyev, Nov 06 2022

			From _Petros Hadjicostas_, Dec 17 2019: (Start)
For n = 2, the 2*a(2) = 8 n X n matrices M with elements in {0,1,2,3,4} that satisfy MM' mod 5 = I are the following:
(a) those with 1 = det(M) mod 5:
[[1,0],[0,1]]; [[0,4],[1,0]]; [[0,1],[4,0]]; [[4,0],[0,4]].
These form the abelian group SO(2, Z_5). See the comments for sequence A060968.
(b) those with 4 = det(M) mod 5:
[[0,1],[1,0]]; [[0,4],[4,0]]; [[1,0],[0,4]]; [[4,0],[0,1]].
Note that, for n = 3, we have 2*a(3) = 2*120 = 240 = A264083(5). (End)

Jessie MacWilliams, Orthogonal Matrices Over Finite Fields, The American Mathematical Monthly 76:2 (1969), 152-164.
Jianing Song, Structure of the group SO(2,Z_n).

Cf. A060968, A071302, A071303, A071305, A071306, A071307, A071308, A071309, A071310, A071900, A087784, A208895, A264083, A318609, A330607.

{ a071304(n) = my(t=n\2); prod(i=0, t-1, 5^(2*t)-5^(2*i)) * if(n%2, 5^t, 1/(5^t+1)); } \\ Max Alekseyev, Nov 06 2022

a(2k+1) = 5^k * Product_{i=0..k-1} (5^(2k) - 5^(2i)); a(2k) = (5^k - 1) * Product_{i=1..k-1} (5^(2k) - 5^(2i)) (see MacWilliams, 1969). - Max Alekseyev, Nov 06 2022
From Petros Hadjicostas, Dec 20 2019: (Start)
Let b(n) be the number of solutions to the equation Sum_{i = 1..n} x_i^2 = 1 (mod 5) with x_i in 0..4. We have that b(n) = 5*b(n-1) + 5*b(n-2) - 25*b(n-3) for n >= 3 with b(0) = 0, b(1) = 2, and b(2) = 4.
We have b(n) = A330607(n, k=1) for n >= 0.
We conjecture that a(n+1) = a(n)*b(n+1) for n >= 1. (End)

Terms a(7) onward from Max Alekseyev, Nov 06 2022

A071306 a(n) = (1/2) * (number of n X n 0..6 matrices M with MM' mod 7 = I, where M' is the transpose of M and I is the n X n identity matrix).

Original entry on oeis.org

1, 8, 336, 112896, 276595200, 4662288691200, 546914437209907200, 450219964711195607040000, 2596509480922336727312302080000, 104784757384177668346109081238896640000, 29597339316082819652234687848790174733434880000

Offset: 1

R. H. Hardin, Jun 11 2002

nonn

Also, number of n X n orthogonal matrices over GF(7) with determinant 1. - Max Alekseyev, Nov 06 2022

			From _Petros Hadjicostas_, Dec 19 2019: (Start)
For n = 2, the 2*a(2) = 16 n X n matrices M with elements in 0..6 that satisfy MM' = I are the following:
(a) those with 1 = det(M) mod 7:
[[1,0],[0,1]]; [[0,1],[6,0]]; [[0,6],[1,0]]; [[2,2],[5,2]];
[[2,5],[2,2]]; [[5,2],[5,5]]; [[5,5],[2,5]]; [[6,0],[0,6]].
These are the elements of the abelian group SO(2,Z_7). See the comments for sequence A060968.
(b) those with 6 = det(M) mod 7:
[[0,1],[1,0]]; [[0,6],[6,0]]; [[1,0],[0,6]]; [[2,2],[2,5]];
[[2,5],[5,5]]; [[5,2],[2,2]]; [[5,5],[5,2]]; [[6,0],[0,1]].
Note that, for n = 3, we have 2*a(3) = 2*336 = 672 = A264083(7). (End)

Jessie MacWilliams, Orthogonal Matrices Over Finite Fields, The American Mathematical Monthly 76:2 (1969), 152-164.

Cf. A060968, A071302, A071303, A071305, A071306, A071307, A071308, A071309, A071310, A071900, A087784, A208895, A264083, A318609.

{ a071306(n) = my(t=n\2); prod(i=0, t-1, 7^(2*t)-7^(2*i)) * if(n%2, 7^t, 1/(7^t+(-1)^t)); } \\ Max Alekseyev, Nov 06 2022

a(2k+1) = 7^k * Product_{i=0..k-1} (7^(2k) - 7^(2i)); a(2k) = (7^k + (-1)^(k+1)) * Product_{i=1..k-1} (7^(2k) - 7^(2i)) (see MacWilliams, 1969). - Max Alekseyev, Nov 06 2022

Conjecture: Let b(n) be the number of solutions to the equation Sum_{i = 1..n} x_i^2 = 1 (mod 7) with x_i in 0..6. We conjecture that b(n) = 7*b(n-1) - 7*b(n-2) + 49*b(n-3) for n >= 4 with b(1) = 2, b(2) = 8, and b(3) = 42. We also conjecture that a(n+1) = a(n)*b(n+1) for n >= 1. - Petros Hadjicostas, Dec 19 2019

Terms a(6) onward from Max Alekseyev, Nov 06 2022

A264083 Number of orthogonal 3 X 3 matrices over the ring Z/nZ.

Original entry on oeis.org

1, 6, 48, 384, 240, 288, 672, 6144, 1296, 1440, 2640, 18432, 4368, 4032, 11520, 49152, 9792, 7776, 13680, 92160, 32256, 15840, 24288, 294912, 30000, 26208, 34992, 258048, 48720, 69120, 59520, 393216, 126720, 58752, 161280, 497664, 101232, 82080, 209664, 1474560

Offset: 1

Charles Repizo, Nov 03 2015

Number of matrices M = [a,b,c; d,e,f; g,h,i] with 0 <= a, b, c, d, e, f, g, h, i < n such that M*transpose(M) == [1,0,0; 0,1,0; 0,0,1] (mod n).

For n > 1, a(n) is divisible by 6*A060594(n)^3. - Robert Israel, Dec 16 2015

Robert Israel, Table of n, a(n) for n = 1..124

Cf. A060594, A087784, A208895.

Enter R := IntegerRing(n);
korthmat := function(R,n,k);
O := [];
M := MatrixAlgebra(R,n);
for x in M do
if x*Transpose(x) eq k*M!1 and Transpose(x)*x eq k*M!1 then
O := Append(O,x);
end if;
end for;
return O;
end function;
# korthmat(R,3,1);

F:= proc(n) local R,V,nR,S,nS,Rp,nRp,i,j,a,b,c,t,r,r1,count;
      R:= select(t -> t[1]^2 + t[2]^2 + t[3]^2 mod n = 1, [seq(seq(seq([a,b,c],a=0..n-1),b=0..n-1),c=0..n-1)]);
      nR:= nops(R);
      S:= select(t -> t^2 mod n = 1, {$2..n-1});
      nS:= nops(S);
      for r in R do if not assigned(V[r]) then
         for c in S do V[c*r mod n] := 0 od
      fi od;
      R:= select(r -> not assigned(V[r]), R);
      nR:= nops(R);
      count:= 0;
      for i from 1 to nR do
        r:= R[i];
        Rp:= select(j -> R[j][1]*r[1] + R[j][2]*r[2] + R[j][3]*r[3] mod n = 0, [$i+1..nR]);
        nRp:= nops(Rp);
        for j from 1 to nRp do
            r1:= R[Rp[j]];
            count:= count + 6*(1+nS)^3*nops(select(k -> R[Rp[k]][1]*r1[1] + R[Rp[k]][2]*r1[2]+R[Rp[k]][3]*r1[3] mod n = 0, [$j+1..nRp]));
        od
      od;
      count;
end proc:
F(1):= 1:
seq(F(n), n=1..40); # Robert Israel, Dec 16 2015

my(t=Mod(matid(3), n)); sum(a=1, n, sum(b=1, n, sum(c=1, n, sum(d=1, n, sum(e=1, n, sum(f=1, n, sum(g=1, n, sum(h=1, n, sum(i=1, n, my(M=[a, b, c; d, e, f; g, h, i]); M*M~==t))))))))) \\ Charles R Greathouse IV, Nov 10 2015

For p an odd prime, a(p) = 2*p*(p^2-1). - Tom Edgar, Nov 04 2015
From Robert Israel, Dec 16 2015: (Start)
Conjectures:
  a(2^k) = 12*8^k for k >= 3.
  For odd primes p, a(p^k) = a(p)*p^(3k-3) for k>=1. (End)

a(11)-a(31) from Tom Edgar, Nov 05 2015

a(31) corrected by Robert Israel, Dec 15 2015

A071900 1/4 times the number of n X n 0..7 matrices with MM' mod 8 = I, where M' is the transpose of M and I is the n X n identity matrix.

Original entry on oeis.org

1, 16, 1536, 786432, 2013265920

Offset: 1

R. H. Hardin, Jun 12 2002

			From _Petros Hadjicostas_, Dec 18 2019: (Start)
For n = 2, the 4*a(2) = 64 n X n matrices M with elements in 0..7 that satisfy MM' mod 8 = I can be classified into four categories:
(a) Matrices M with 1 = det(M) mod 8. These form the abelian group SO(2, Z_8). See the comments for sequence A060968.
(b) Matrices M with 3 = det(M) mod 8. These are the elements of the left coset A*SO(2, Z_8) = {AM: M in SO(2, Z_8)}, where A = [[3,0],[0,1]].
(c) Matrices M with 5 = det(M) mod 8. These are the elements of the left coset B*SO(2, Z_8) = {BM: M in SO(2, Z_8)}, where B = [[5,0],[0,1]].
(d) Matrices M with 7 = det(M) mod 8. These are the elements of the left coset C*SO(2, Z_8) = {CM: M in SO(2, Z_8)}, where C= [[7,0],[0,1]].
All four classes of matrices have the same number of elements, that is, 16 each.
Note that for n = 3 we have 4*a(3) = 4*1536 = 6144 = A264083(8). (End)

Jianing Song, Structure of the group SO(2,Z_n).
László Tóth, Counting solutions of quadratic congruences in several variables revisited, arXiv:1404.4214 [math.NT], 2014.
László Tóth, Counting Solutions of Quadratic Congruences in Several Variables Revisited, J. Int. Seq. 17 (2014), #14.11.6.

Cf. A060968, A071302, A071303, A071304, A071305, A071306, A071307, A071308, A071309, A071310, A071900, A087784, A208895, A264083.

Conjecture: a(n) = 2^(n*(n-1)/2) * A071303(n) for n >= 1. - Michel Marcus, Nov 08 2022

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

A227553 Number of solutions to x^2 - y^2 - z^2 == 1 (mod n).

Original entry on oeis.org

1, 4, 6, 8, 30, 24, 42, 32, 54, 120, 110, 48, 182, 168, 180, 128, 306, 216, 342, 240, 252, 440, 506, 192, 750, 728, 486, 336, 870, 720, 930, 512, 660, 1224, 1260, 432, 1406, 1368, 1092, 960, 1722, 1008, 1806, 880, 1620, 2024, 2162, 768, 2058, 3000, 1836

Offset: 1

José María Grau Ribas, Jul 16 2013

Conjecture: a(2) = 4; if s > 1 then a(2^s) = 2^(2s-1); if p == 1 (mod 4) then a(p^s) = (p+1)*p^(2s-1); if p == 3 (mod 4) then a(p^s) = (p-1)*p^(2s-1).

Andrew Howroyd, Table of n, a(n) for n = 1..2500
L. Toth, Counting Solutions of Quadratic Congruences in Several Variables Revisited, J. Int. Seq. 17 (2014) # 14.11.6.

Cf. A208895, A086932, A089003, A060968, A087784.

a[1] = 1; a[n_] := Sum[If[Mod[a^2-b^2-c^2, n] == 1, 1, 0], {a, n}, {b, n}, {c, n}]; Table[a[n], {n, 10}]

M(n,f)={sum(i=0, n-1, Mod(x^(f(i)%n), x^n-1))}
a(n)={polcoeff(lift(M(n, i->i^2) * M(n, i->-(i^2))^2 ), 1%n)} \\ Andrew Howroyd, Jun 24 2018

cp's OEIS Frontend

A087784 Number of solutions to x^2 + y^2 + z^2 = 1 mod n.

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A071302 a(n) = (1/2) * (number of n X n 0..2 matrices M with MM' mod 3 = I, where M' is the transpose of M and I is the n X n identity matrix).

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A071303 1/2 times the number of n X n 0..3 matrices M with MM' mod 4 = I, where M' is the transpose of M and I is the n X n identity matrix.

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A071304 a(n) = (1/2) * (number of n X n 0..4 matrices M with MM' mod 5 = I, where M' is the transpose of M and I is the n X n identity matrix).

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PARI

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Extensions

A071306 a(n) = (1/2) * (number of n X n 0..6 matrices M with MM' mod 7 = I, where M' is the transpose of M and I is the n X n identity matrix).

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Programs

PARI

Formula

Extensions

A264083 Number of orthogonal 3 X 3 matrices over the ring Z/nZ.

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Magma

Maple

PARI

Formula

Extensions

A071900 1/4 times the number of n X n 0..7 matrices with MM' mod 8 = I, where M' is the transpose of M and I is the n X n identity matrix.

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Formula

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