A052182 -id:A052182 - cp's OEIS frontend

A383774 a(n) = neg(M(n)), where M(n) is the n X n left circulant matrix with (row 1) = (1, 2, ... , n), and neg(M(n)) is the negative part of the determinant of M(n); see A380661.

0, -4, -36, -450, -15675, -772122, -47753440, -3909436192, -410168886615, -53329052728000, -8417606908865220, -1586195621597483136, -351735180091505203539, -90667510133054591492224, -26884188980472806091900000, -9086147124746080046118543360

Offset: 1

Clark Kimberling, May 17 2025

sign

			The rows of M(4) are (1, 2, 3, 4), (2, 3, 4, 1), (3, 4, 1, 2), (4, 1, 2, 3); determinant(M(4)) = 160; permanent(M(4)) = 1060, so neg(M(4)) = (160 - 1060)/2 = -450 and pos(M(4)) = (160 + 1060)/2 = 610.

Cf. A052182 (determinant), A085719 (permanent), A380661, A383772, A383773, A383775.

z = 18;
v[n_] := Table[k + 1, {k, 0, n - 1}];
u[n_] := Table[RotateLeft[#, k - 1], {k, 1, Length[#]}] &[v[n]];
p = Table[Simplify[Permanent[u[n]]], {n, 1, z}]   (* A085719 *)
d = Table[Simplify[Det[u[n]]], {n, 1, z}] (* A052182, with altered signs *)
neg = (d - p)/2   (* A383774 *)
pos = (d + p)/2   (* A383775 *)

A383775 a(n) = pos(M(n)), where M(n) is the n X n left circulant matrix with (row 1) = (1, 2, ... , n), and pos(M(n)) is the positive part of the determinant of M(n); see A380661.

Original entry on oeis.org

1, 1, 18, 610, 17550, 744906, 47282844, 3918873376, 410384120220, 53323552728000, 8417451284317614, 1586200451151892608, 351735343178101060906, 90667504180193792086144, 26884188746929397888775000, 9086147134545912835276742656, 3472279409772212369077001352888

Offset: 1

Clark Kimberling, May 22 2025

nonn

			The rows of M(4) are (1, 2, 3, 4), (2, 3, 4, 1), (3, 4, 1, 2), (4, 1, 2, 3); determinant(M(4)) = 160; permanent(M(4)) = 1060, so neg(M(4)) = (160 - 1060)/2 = -450 and pos(M(4)) = (160 + 1060)/2 = 610.

Cf. A052182 (determinant), A085719 (permanent), A380661, A383772, A383773, A383774.

z = 18;
v[n_] := Table[k + 1, {k, 0, n - 1}];
u[n_] := Table[RotateLeft[#, k - 1], {k, 1, Length[#]}] &[v[n]];
p = Table[Simplify[Permanent[u[n]]], {n, 1, z}]   (* A085719 *)
d = Table[Simplify[Det[u[n]]], {n, 1, z}] (* A052182, with altered signs *)
neg = (d - p)/2   (* A383774 *)
pos = (d + p)/2   (* A383775 *)

A079340 Absolute value of determinant of n X n matrix whose entries are the integers from 1 to n^2 spiraling outward, ending in a corner.

Original entry on oeis.org

1, 5, 72, 1380, 31920, 861840, 26611200, 925404480, 35805369600, 1526139014400, 71066912716800, 3590219977344000, 195589552648089600, 11430978821982720000, 713448513897799680000, 47363888351558338560000

Offset: 1

Kit Vongmahadlek (kit119(AT)yahoo.com), Jan 03 2003

If n == 0 or 1 (mod 4), the sign of the determinant will be independent of the orientation of the spiral. For n == 2 or 3 (mod 4), the sign will be reversed when the orientation is rotated by 1/4 or flipped on the horizontal or vertical axis. - Franklin T. Adams-Watters, Dec 31 2013

This distribution of the integers is sometimes known as Ulam's spiral, although that is sometimes reserved for when the primes are marked out in some way. - Franklin T. Adams-Watters, Dec 31 2013

			n=2, det=-5: {1 2 / 4 3 }.
n=3, det=72: {7 8 9 / 6 1 2 / 5 4 3 }.
n=4, det=-1380: { 7 8 9 10 / 6 1 2 11 /5 4 3 12 / 16 15 14 13 }.
n=5, det=31920: { 21 22 23 24 25 / 20 7 8 9 10 / 19 6 1 2 11 /18 5 4 3 12 / 17 16 15 14 13 }

Gaurav Bhatnagar, Christian Krattenthaler, Spiral determinants, arXiv:1704.02859 [math.CO], 2017.

Cf. A078475, A023999, A067276, A052182.

M[0, 0] = 1;
M[i_, j_] := If[i <= j,
  If[i + j >= 0, If[i != j, M[i + 1, j] + 1, M[i, j - 1] + 1],
   M[i, j + 1] + 1],
  If[i + j > 1, M[i, j - 1] + 1, M[i - 1, j] + 1]
  ]
M[n_] := If[EvenQ[n],
  Table[M[i, j], {j, n/2, -n/2 + 1, -1}, {i, -n/2 + 1, n/2}],
  Table[M[i, j], {j, (n - 1)/2, -(n - 1)/2, -1}, {i, -(n - 1)/2, (n - 1)/2}]]
a[n_]:=Det[M[n]] (* Christian Krattenthaler, Apr 19 2017 *)

A079340(n):=if n=1 then 1 else (2*n^2-3*n+3)*(2*n-2)!/(2*(n-1)!)$
makelist(A079340(n),n,1,30); /* Martin Ettl, Nov 05 2012 */

a(n) = (2*n^2-3*n+3) (2n-2)!/(2 (n-1)!) = A096376(n-1)*A000407(n-2), n>1. - Conjectured by Dean Hickerson, Jan 30 2003. Proved in the article by Bhatnagar and Krattenthaler.

D-finite with recurrence (2*n^2-7*n+8)*a(n) -2*(2*n-3)*(2*n^2-3*n+3)*a(n-1)=0. - R. J. Mathar, May 03 2019

Extended by Robert G. Wilson v, Jan 25 2003

A118704 a(n) = determinant of n X n circulant matrix whose first row is the first n distinct Fibonacci numbers A000045(2), A000045(3), ... A000045(n+1).

Original entry on oeis.org

1, -3, 18, -429, 24149, -3813376, 1513739413, -1575456727131, 4215561680804992, -29321025953223722025, 529210578655758192641625, -24875949855198086445567836160, 3047957640551011125902187378426905, -974921913036976554924444728974464589255

Offset: 1

Jonathan Vos Post, May 20 2006

a(n) alternates in sign.

			a(2) = -3 because of the determinant -3 =
| 1, 2 |
| 2, 1 |.
a(5) = 24149 = determinant
| 1, 2, 3, 5, 8 |
| 8, 1, 2, 3, 5 |
| 5, 8, 1, 2, 3 |
| 3, 5, 8, 1, 2 |
| 2, 3, 5, 8, 1 |.

Alois P. Heinz, Table of n, a(n) for n = 1..50
Eric Weisstein's World of Mathematics, Circulant Matrix.

See also: A048954 Wendt determinant of n-th circulant matrix C(n). A052182 Circulant of natural numbers. A066933 Circulant of prime numbers. A086459 Circulant of powers of 2.

Cf. A000045, A048954, A052182, A066933, A086459, A086569.

a:= n-> LinearAlgebra[Determinant](Matrix(n, (i, j)->
        (<<0|1>, <1|1>>^(2+irem(n-i+j, n)))[1, 2])):
seq(a(n), n=1..15);  # Alois P. Heinz, Oct 23 2009

a(n) ~ (-1)^(n+1) * phi^(n*(n+1)) / 5^(n/2), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jul 10 2025

Corrected and extended by Alois P. Heinz, Oct 23 2009

A118705 a(n) = determinant of n X n circulant matrix whose first row is the first n triangular numbers A000217(0), A000217(1), ... A000217(n-1).

Original entry on oeis.org

0, -1, 28, -1360, 105500, -12051585, 1908871832, -400855203840, 107838796034520, -36175347978515625, 14806446317943766420, -7263073394295238840320, 4206546078973080241293076, -2840250692354398785860048105, 2211476237421629752792968750000

Offset: 1

Jonathan Vos Post, May 20 2006

			a(2) = - 1 because of the determinant -1 =
  | 0, 1 |
  | 1, 0 |.
a(4) = -1360 = determinant
  |0,1,3,6|
  |6,0,1,3|
  |3,6,0,1|
  |1,3,6,0|.

Robert Israel, Table of n, a(n) for n = 1..226
Eric Weisstein's World of Mathematics, Circulant Matrix.

See also: A048954 Wendt determinant of n-th circulant matrix C(n). A052182 Circulant of natural numbers. A066933 Circulant of prime numbers. A086459 Circulant of powers of 2.

Cf. A000045, A048954, A052182, A066933, A086459, A086569.

f:= proc(n) uses LinearAlgebra;local i;
  Determinant(Matrix(n, shape=Circulant[[seq(i*(i+1)/2, i=0..n-1)]]))
end proc:
map(f, [$1..30]); # Robert Israel, Jan 25 2023

r[n_] := r[n] = Table[k(k+1)/2, {k, 0, n-1}];
M[n_] := Table[RotateRight[r[n], m-1], {m, 1, n}];
a[n_] := Det[M[n]];
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Feb 11 2023 *)

a(n) = (-1)^(n-1)*n^(n-2)*(n+1)*(n-1)*((n+1)^n-(n-1)^n)/(6*2^n). [Missouri State University Problem-Solving Group (MSUPSG(AT)MissouriState.edu), May 03 2010]

More terms from Alois P. Heinz, Mar 16 2017

A076863 n^(n-2)+(n(n-1)/2)(n-1)^(n-3).

Original entry on oeis.org

1, 2, 6, 34, 285, 3171, 44023, 732740, 14220153, 315233605, 7857947691, 217541911830, 6621714803509, 219780508232663, 7899055867765455, 305601002241052936, 12662255840668015089, 559390257977596068105, 26246546675302419239635, 1303417502979112415328410

Offset: 1

N. J. A. Sloane, Nov 23 2002

nonn

For n > 4, number of maximal independent generating sets for S_n. Compare A078103.

P. J. Cameron and P. Cara, Independent generating sets and geometries for symmetric groups, J. Algebra, Vol. 258, no. 2 (2002), 641-650.

A000272 + A052182.

Join[{1},Table[n^(n-2)+(n(n-1))/2 (n-1)^(n-3),{n,2,20}]] (* Harvey P. Dale, Apr 26 2017 *)

A118707 a(n) = determinant of n X n circulant matrix whose first row is the first n square numbers 0, 1, ..., (n-1)^2.

Original entry on oeis.org

0, -1, 65, -6720, 1080750, -252806400, 81433562119, -34630270976000, 18813448225370124, -12719917900800000000, 10478214213011739186685, -10333870908014534470926336, 12023263324381930168836397850, -16297888825404790818315505238016

Offset: 1

Jonathan Vos Post, May 20 2006

			a(2) = -1 because of the determinant -1 =
| 0, 1 |
| 1, 0 |.
a(3) = 65 = determinant
|0,1,4|
|4,0,1|
|1,4,0|.

Eric Weisstein's World of Mathematics, Circulant Matrix.

See also: A000290 The squares: a(n) = n^2. A048954 Wendt determinant of n-th circulant matrix C(n). A052182 Circulant of natural numbers. A066933 Circulant of prime numbers. A086459 Circulant of powers of 2.

Cf. A000290, A048954, A052182, A066933, A086459, A086569.

a(n) = (-1)^(n-1)*(n-1)*(2*n-1)*n^(n-2)*(n^n-(n-2)^n)/12 [From Missouri State University Problem-Solving Group (MSUPSG(AT)MissouriState.edu), May 05 2010]

More terms from Alois P. Heinz, Mar 16 2017

A118709 a(n) = determinant of n X n circulant matrix whose first row is the first n cube numbers 0, 1, ..., (n-1)^3.

Original entry on oeis.org

0, -1, 513, -532800, 1077540500, -3831689610000, 22051842087895137, -192710430555501494272, 2433436736207275231050384, -42684202683959414242500000000, 1007311823853329619224620155226025, -31149342348518897782279760206406615040

Offset: 1

Jonathan Vos Post, May 20 2006

			a(2) = -1 because of the determinant -1 =
| 0, 1 |
| 1, 0 |.
a(3) = 513 = determinant
|0,1,8|
|8,0,1|
|1,8,0|.
a(6) = 22051842087895137 = determinant
|0,1,8,27,64,125,216|
|216,0,1,8,27,64,125|
|125,216,0,1,8,27,64|
|64,125,216,0,1,8,27|
|27,64,125,216,0,1,8|
|8,27,64,125,216,0,1|
|1,8,27,64,125,216,0|.

Eric Weisstein's World of Mathematics, Circulant Matrix.

See also: A000578 The cubes: a(n) = n^3. A048954 Wendt determinant of n-th circulant matrix C(n). A052182 Circulant of natural numbers. A066933 Circulant of prime numbers. A086459 Circulant of powers of 2.

Cf. A000578, A048954, A052182, A066933, A086459, A086569.

Table[Det[Table[RotateRight[Range[0,i]^3,n],{n,0,i}]],{i,0,10}] (* Harvey P. Dale, Oct 22 2012 *)

Contribution from Missouri State University Problem-Solving Group (MSUPSG(AT)MissouriState.edu), May 05 2010: (Start)
a(n) = (-1)^(n-1)*(n-1)^2*n^(n-2)*(n^(2n)-b(n)^n-c(n)^n+(n^2-3n+3)^n)/24
where
b(n)=(2*n^2-3*n-3+sqrt(15n^2-18n-9)i)/2 and
c(n)=(2*n^2-3*n-3-sqrt(15n^2-18n-9)i)/2 (End)

More terms from Harvey P. Dale, Oct 22 2012

A118712 a(n) = Determinant of n X n circulant matrix whose first row is A000001(1), A000001(2), ..., A000001(n) where A000001(n) = number of groups of order n.

Original entry on oeis.org

1, 0, 0, -5, 6, -16, 9, -134400, 647248, -1711908, 6076067, -85248000, 116477425, -1764364437, 909276004, -522319050599375232, 14313181351994538493, -165893335414907083200, 2939566160282258664451, -5007637771411479278976, 75399747694572065660672

Offset: 1

Jonathan Vos Post, May 20 2006

sign

			a(4) = -5 because of the determinant -5 =
|1,1,1,2|
|2,1,1,1|
|1,2,1,1|
|1,1,2,1|.
a(11) = 6076067 = determinant
|1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1|
|1, 1, 1, 1, 2, 1, 2, 1, 5, 2, 2|
|2, 1, 1, 1, 1, 2, 1, 2, 1, 5, 2|
|2, 2, 1, 1, 1, 1, 2, 1, 2, 1, 5|
|5, 2, 2, 1, 1, 1, 1, 2, 1, 2, 1|
|1, 5, 2, 2, 1, 1, 1, 1, 2, 1, 2|
|2, 1, 5, 2, 2, 1, 1, 1, 1, 2, 1|
|1, 2, 1, 5, 2, 2, 1, 1, 1, 1, 2|
|2, 1, 2, 1, 5, 2, 2, 1, 1, 1, 1|
|1, 2, 1, 2, 1, 5, 2, 2, 1, 1, 1|
|1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 1|.

Eric Weisstein's World of Mathematics, Circulant Matrix.

Cf. A000001, A048954, A052182, A066933, A086459, A086569.

A118712 := n -> DeterminantMat(List([0..n-1], i->List([0..n-1], j->NrSmallGroups(((j-i) mod n)+1)))); # Eric M. Schmidt, Nov 17 2013

a(1) corrected by and more terms from Eric M. Schmidt, Nov 17 2013

A338424 Permanent of n X n matrix whose rows are cyclic permutations of 2..prime(n).

Original entry on oeis.org

1, 2, 13, 250, 8961, 821848, 90031117, 16929468082, 3699112438993, 1172372082948016, 592930150554224105, 314980461163972949868, 237098727748233796765441, 209617206588771062279193558, 193794661072190257001413994605, 210096924743130844484817519143524, 287845708921326527700201694718406369

Offset: 0

Stefano Spezia, Oct 25 2020

nonn

			a(3) = 250 because the permanent of [(2,3,5), (3,5,2), (5,2,3)] is 250.

Vaclav Kotesovec, Table of n, a(n) for n = 0..36

Cf. A000040, A052182, A066933, A070896, A086759.

a(n) = matpermanent(matrix(n, n, i, j, prime((i+j-2) % n + 1)))

cp's OEIS Frontend

A383774 a(n) = neg(M(n)), where M(n) is the n X n left circulant matrix with (row 1) = (1, 2, ... , n), and neg(M(n)) is the negative part of the determinant of M(n); see A380661.

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A383775 a(n) = pos(M(n)), where M(n) is the n X n left circulant matrix with (row 1) = (1, 2, ... , n), and pos(M(n)) is the positive part of the determinant of M(n); see A380661.

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A079340 Absolute value of determinant of n X n matrix whose entries are the integers from 1 to n^2 spiraling outward, ending in a corner.

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A118704 a(n) = determinant of n X n circulant matrix whose first row is the first n distinct Fibonacci numbers A000045(2), A000045(3), ... A000045(n+1).

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A118705 a(n) = determinant of n X n circulant matrix whose first row is the first n triangular numbers A000217(0), A000217(1), ... A000217(n-1).

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A076863 n^(n-2)+(n*(n-1)/2)*(n-1)^(n-3).

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A118707 a(n) = determinant of n X n circulant matrix whose first row is the first n square numbers 0, 1, ..., (n-1)^2.

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A118709 a(n) = determinant of n X n circulant matrix whose first row is the first n cube numbers 0, 1, ..., (n-1)^3.

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A076863 n^(n-2)+(n(n-1)/2)(n-1)^(n-3).