A052182 -id:A052182 - cp's OEIS frontend

A048954 Wendt determinant of n-th circulant matrix C(n).

1, -3, 28, -375, 3751, 0, 6835648, -1343091375, 364668913756, -210736858987743, 101832157445630503, 0, 487627751563388801409591, -4875797582053878382039400448, 58623274842128064372315087290368, -1562716604740038367719196682456673375

Offset: 1

Eric W. Weisstein

det(C(n)) = 0 for n divisible by 6.
The determinant of the circulant matrix is 0 when 6 divides n because the polynomial (x+1)^(6k) - 1 has roots that are roots of unity. See A086569 for a generalization. - T. D. Noe, Jul 21 2003
E. Lehmer claimed and J. S. Frame proved that 2^n - 1 divides a(n) and the quotient abs(a(n))/(2^n - 1) is a perfect square (Ribenboim 1999, p. 128). - Jonathan Sondow, Aug 17 2012
C(n) is the matrix whose first row is [c_1, ..., c_n] where c_i = binomial(n,i-1), and subsequent rows are obtained by cyclically shifting the previous row one place to the right: see examples and PARI code. - M. F. Hasler, Dec 17 2016

			a(2) = det [ 1 2 ; 2 1 ] = -3.
a(3) = det [ 1 3 3 ; 3 1 3 ; 3 3 1 ] = 28.
a(4) = det [ 1 4 6 4 ; 4 1 4 6 ; 6 4 1 4 ; 4 6 4 1 ] = -375.

P. Ribenboim, "Fermat's Last Theorem for Amateurs", Springer-Verlag, NY, 1999, pp. 126, 136.
P. Ribenboim, 13 Lectures on Fermat's last theorem, Springer-Verlag, NY, 1979, pp. 61-63. MR0551363 (81f:10023).

T. D. Noe, Table of n, a(n) for n=1..50
David W. Boyd, The asymptotic behaviour of the binomial circulant determinant, Journal of Mathematical Analysis and Applications, Volume 86, Issue 1, March 1982, Pages 30-38.
E. Brown and M. Chamberland, Generalizing Gauss's Gem, Amer. Math. Monthly, 119 (No. 7, 2012), 597-601. - _N. J. A. Sloane_, Sep 07 2012
D. Burde and W. A. Moens, The structure of Lie algebras with a derivation satisfying a polynomial identity, arXiv:2009.05434 [math.RA], 2020.
L. Carlitz, A determinant connected with Fermat's last theorem, Proc. Amer. Math. Soc. 10 (1959), 686-690.
L. Carlitz, A determinant connected with Fermat's last theorem, Proc. Amer. Math. Soc. 11 (1960), 730-733.
Joshua Cooper and Zhibin Du, Note on the spectra of Steiner distance hypermatrices, arXiv:2403.02287 [math.CO], 2024. See pp. 2, 4.
Greg Fee and Andrew Granville, The prime factors of Wendt's binomial circulant determinant, Math. Comp. 57 (1991), 839-848.
David Ford and Vijay Jha, On Wendt's Determinant and Sophie Germain's Theorem, Experimental Mathematics, 2 (1993) No. 2, 113-120.
J. S. Frame, Factors of the binomial circulant determinant, Fibonacci Quart., 18 (1980), pp. 9-23.
Charles Helou, On Wendt's Determinant, Math. Comp., 66 (1997) No. 219, 1341-1346.
Charles Helou and Guy Terjanian, Arithmetical properties of wendt's determinant, Journal of Number Theory, Volume 115, Issue 1, November 2005, Pages 45-57.
Emma Lehmer, On a resultant connected with Fermat's last theorem, Bull. Amer. Math. Soc. 41 (1935), 864-867.
Gerard P. Michon, Factorization of Wendt's Determinant (table for n=1 to 114)
Anastasios Simalarides, Upper bounds for the prime divisors of Wendt's determinant, Math. Comp., 71 (2002), 415-427.
Eric Weisstein's World of Mathematics, Circulant matrix
E. Wendt, Arithmetische Studien über den letzten Fermatschen Satz, welcher aussagt, dass die Gleichung a^n=b^n+c^n für n>2 in ganzen Zahlen nicht auflösbar ist, Reimer (Berlin), 1894.

Cf. A052182 (circulant of natural numbers), A066933 (circulant of prime numbers), A086459 (circulant of powers of 2), A086569, A129205, A215615, A215616.

See A096964 for another definition.

a[ n_] := Resultant[ x^n - 1, (1+x)^n - 1, x];

{a(n) = if( n<1, 0, matdet( matrix( n, n, i, j, binomial( n, (j-i)%n ))))}

a(n) = polresultant( x^n - 1, (1+x)^n - 1, x )

a(2*n) = A129205(n)^2 * (1-4^n).
a(n) = 0 if and only if 6 divides n. If d divides n, then a(d) divides a(n). - Michael Somos, Apr 03 2007
a(n) = (-1)^(n-1) * (2^n - 1) * A215615(n)^2. - Jonathan Sondow, Aug 17 2012
a(2*n) = -3 * A215616(n)^3. - Jonathan Sondow, Aug 18 2012

Additional comments from Michael Somos, May 27 2000 and Dec 16 2001

A066933 Determinant of n X n matrix whose rows are cyclic permutations of 2..prime(n).

Original entry on oeis.org

1, 2, -5, -70, 1275, 97748, -2713585, -251983958, 9651414311, 1137214908700, -268100912462097, -16553358418854560, 4303513869962179379, 602501593820064477686, -50199332236439321779977, -7847812115804566640572424, 2754406130856424049914030863

Offset: 0

Robert G. Wilson v, Jan 24 2002

			a(3) = -70 because this is the determinant of [(2,3,5), (3,5,2), (5,2,3)].

Alois P. Heinz, Table of n, a(n) for n = 0..300

Cf. A052182.

a:= n-> LinearAlgebra[Determinant](Matrix(n,
       (i, j)-> ithprime(1+irem(i+j-2, n)))):
seq(a(n), n=0..20);  # Alois P. Heinz, Dec 09 2016

f[ n_ ] := Module[ {a = Table[ Prime[ i ], {i, 1, n} ], m = {}, k = 0}, While[ k < n, m = Append[ m, RotateLeft[ a, k ] ]; k++ ]; Det[ m ] ]; Table[ f[ n ], {n, 1, 16} ]

a(n) = matdet(matrix(n, n, i, j, prime(1+lift(Mod(i+j-2, n))))); \\ Michel Marcus, Aug 11 2019; corrected Jun 12 2022

Conjecture: a(n) = (-1)^(n+floor(n/2))*Res(f(n) , x^n - 1), where Res is the resultant, and f(n)=Sum_{k=1..n} prime(k)*x^k. - Benedict W. J. Irwin, Dec 07 2016

a(0)=1 prepended by Alois P. Heinz, Dec 09 2016

A086459 Determinant of the circulant matrix whose rows are formed by successively rotating the vector (1, 2, 4, 8, ..., 2^(n-1)) right.

Original entry on oeis.org

1, -3, 49, -3375, 923521, -992436543, 4195872914689, -70110209207109375, 4649081944211090042881, -1227102111503512992112190463, 1291749870339606615892191271170049, -5429914198235566686555216227881787109375

Offset: 1

T. D. Noe, Jul 21 2003

Note that if the rows are rotated left instead of right, the sign of the terms for which n = 0 or 3 (mod 4) is reversed. The n eigenvalues of these circulant matrices lie on the circle of radius 2(2^n - 1)/3 centered at x = (2^n - 1)/3, y = 0. This sequence can be generalized to bases other than 2 and similar results are true.

			a(3) = determinant of the matrix ((1,2,4),(4,1,2),(2,4,1)) = 49. [Corrected by _T. D. Noe_, Jan 22 2008]

Richard Bellman, Introduction to Matrix Analysis, Second Edition, SIAM, 1970, pp. 242-3.
Philip J. Davis, Circulant Matrices, Second Edition, Chelsea, 1994.

Eric Weisstein's World of Mathematics, Circulant Matrix

Cf. A048954 (circulant of binomial coefficients), A052182 (circulant of natural numbers), A066933 (circulant of prime numbers).

Cf. A180602 (unsigned, offset 0). [Paul D. Hanna, Sep 11 2010]

restart:with (combinat):a:=n->mul(-stirling2(n,2), j=3..n): seq(a(n), n=2..19); # Zerinvary Lajos, Jan 01 2009

Table[x=2^Range[0, n-1]; m=Table[RotateRight[x, i-1], {i, n}]; Det[m], {n, 12}]

a(n) = (-2^n + 1)^(n-1).

See formulas in A180602, an unsigned version of this sequence with offset 0. [Paul D. Hanna, Sep 11 2010]

A070896 Determinant of the Cayley addition table of Z_{n}.

Original entry on oeis.org

0, -1, -9, 96, 1250, -19440, -352947, 7340032, 172186884, -4500000000, -129687123005, 4086546038784, 139788510734886, -5159146026151936, -204350482177734375, 8646911284551352320, 389289535005334947848, -18580248257778920521728

Offset: 1

Santi Spadaro, May 23 2002

sign

a(n) is the determinant of the n X n matrix M_(i,j) = ((i+j) mod n) where i and j range from 0 to n-1. - Benoit Cloitre, Nov 29 2002

|a(n)| = number of labeled mappings from n points to themselves (endofunctions) with an even number of cycles. E.g.f.: (1/2)*LambertW(-x)^2/(1+LambertW(-x)). - Vladeta Jovovic, Mar 30 2006

			a(3) = -9 because the determinant of {{0,1,2}, {1,2,0}, {2,0,1}} is -9.

G. C. Greubel, Table of n, a(n) for n = 1..385

Cf. A000312, A052182, A060281, A060435.

[(-1)^Floor(n/2)*(1/2)*(n-1)*n^(n-1): n in [1..50]]; // G. C. Greubel, Nov 14 2017

Table[(-1)^Floor[n/2]*(1/2)*(n - 1)*n^(n - 1), {n, 1, 50}] (* G. C. Greubel, Nov 14 2017 *)

a(n)=(-1)^floor(n/2)*(1/2)*(n-1)*n^(n-1)

a(n) = (-1)^floor(n/2)*(1/2)*(n-1)*n^(n-1). - Benoit Cloitre, Nov 29 2002

A118713 a(n) = determinant of n X n circulant matrix whose first row is A001358(1), A001358(2), ..., A001358(n) where A001358(n) = n-th semiprime.

Original entry on oeis.org

4, -20, 361, -3567, 218053, -3455872, 736439027, -16245418225, 1519211613654, -37662452460912, 20199655476042865, -643524421698841536, 46513669467992431114, -3754367220494585505280, 277686193779526116536293, -123973821931125256333959105, 20103033234038999233385180658

Offset: 1

Jonathan Vos Post, May 20 2006

Semiprime analog of A066933 Circulant of prime numbers. a(n) alternates in sign. A048954 Wendt determinant of n-th circulant matrix C(n). A052182 Circulant of natural numbers. A086459 Circulant of powers of 2.

			a(2) = -20 = determinant
|4,6|
|6,4|.
a(3) = 361 = 19^2 = determinant
|4,6,9|
|9,4,6|
|6,9,4|.

Eric Weisstein's World of Mathematics, Circulant Matrix.

Cf. A001358, A048954, A052182, A066933, A086459, A086569.

A118713 := proc(n)
    local C,r,c ;
    C := Matrix(1..n,1..n) ;
    for r from 1 to n do
    for c from 1 to n do
        C[r,c] := A001358(1+((c-r) mod n)) ;
    end do:
    end do:
    LinearAlgebra[Determinant](C) ;
end proc:
seq(A118713(n),n=1..13) ;

nmax = 13;
sp = Select[Range[3 nmax], PrimeOmega[#] == 2&];
a[n_] := Module[{M}, M[1] = sp[[1 ;; n]];
   M[k_] := M[k] = RotateRight[M[k - 1]];
   Det[Table[M[k], {k, 1, n}]]];
Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, Feb 16 2023 *)

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Aug 23 2007

A023999 Absolute value of determinant of n X n matrix whose entries are the integers from 1 to n^2 spiraling inward, starting in a corner.

Original entry on oeis.org

1, 5, 48, 660, 11760, 257040, 6652800, 198918720, 6745939200, 255826771200, 10727081164800, 492775291008000, 24610605962342400, 1327677426915840000, 76940526008586240000, 4766815315895592960000, 314406967644177408000000, 21995911456386651463680000

Offset: 1

Charles Diminnie (charles.diminnie(AT)rampo.angelo.edu)

nonn

Starting in the NW or SE corner, the signs are cyclic (+,-,-,+), starting in the NE or SW corner, the signs are always positive.

			n=4: det of
.1..2..3.4
12.13.14.5
11.16.15.6
10..9..8.7

Alois P. Heinz, Table of n, a(n) for n = 1..200
Gaurav Bhatnagar, Christian Krattenthaler, Spiral determinants, arXiv:1704.02859 [math.CO], 2017.
Charles Vanden Eynden, Problem 1517, Mathematics Magazine, Vol. 70, No. 1, Feb., 1997 p. 65.

Cf. A079340, A078475, A067276, A052182.

Main diagonal of A226167, A126224 (signed version). - Alois P. Heinz, Jan 21 2014

a:= proc(n) option remember; `if`(n<2, (3*n+1)/4,
      4*(3*n-1)*(2*n-5)*(2*n-3) *a(n-2) /(3*n-7))
    end:
seq(a(n), n=1..20);  # Alois P. Heinz, Jan 21 2014

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_] := n^2 + 1 - 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 *)

A023999(n):=if n=1 then 1 else 2*((-1)^((n+4)*(n-1))/2 *(3*n-1) * (2*n-3)!/(n-2)!)$
makelist(A023999(n),n,1,30); /* Martin Ettl, Nov 05 2012 */

a(n) = (3n-1) * (2n-3)!/(n-2)! for n >= 2. [corrected by Robert Israel, Apr 20 2017]

E.g.f.: ((-2*x-1)*sqrt(1-4*x)+1-4*x)/(16*x-4). - Robert Israel, Apr 20 2017

Edited and extended by Robert G. Wilson v, May 07 2003

A085719 Permanent of n X n matrix whose rows are cyclic permutations of 1..n.

Original entry on oeis.org

1, 1, 5, 54, 1060, 33225, 1517028, 95036284, 7828309568, 820553006835, 106652605456000, 16835058193182834, 3172396072749375744, 703470523269606264445, 181335014313248383578368, 53768377727402203980675000, 18172294259291992881395286016

Offset: 0

Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 19 2003

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..23 (terms n=1..22 from Herman Jamke)
Wikipedia, Permanent (mathematics)

Cf. A052182, A070896.

a:= n-> `if`(n=0, 1, LinearAlgebra[Permanent](
         Matrix(n, (i, j)-> 1+irem(j+i, n)))):
seq(a(n), n=0..17);  # Alois P. Heinz, Apr 28 2020

permRWNb(a)=n=matsize(a)[1];if(n==1,return(a[1,1]));sg=1;in=vectorv(n);x=in;x=a[,n]-sum(j=1,n,a[,j])/2;p=prod(i=1,n,x[i]);for(k=1,2^(n-1)-1,sg=-sg;j=valuation(k,2)+1;z=1-2*in[j];in[j]+=z;x+=z*a[,j];p+=prod(i=1,n,x[i],sg));return(2*(2*(n%2)-1)*p)
for(n=1,22,a=matrix(n,n,i,j,1+(j-i)%n);print1(permRWNb(a)",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 13 2007

def A085719(n) : return matrix([[(i-j)%n+1 for j in range(n)] for i in range(n)]).permanent() # Eric M. Schmidt, May 04 2013

More terms from Vladeta Jovovic, Jul 21 2003

a(0)=1 prepended by Alois P. Heinz, Apr 28 2020

A123745 Circulants of Fibonacci numbers (without F_0 = 0).

Original entry on oeis.org

1, 0, 4, 35, 1812, 170240, 46301673, 30413016864, 52171354014208, 228072747428273319, 2583414317082067853704, 75732718487930382583857152, 5773860969402842827019263155009, 1146353725688692827225795357533033072, 593830518002528577221255815133242142736384

Offset: 1

Wolfdieter Lang, Nov 10 2006, Jan 27 2009

A circulant C_n is the determinant of a circulant n X n matrix M, i.e. one with entries M_{i,j}=a_{i-j} where the indices are taken mod n. Notation: C_n = C_n([a_n,a_{n-1},...,a_1]), with the first row of M given.
The name circulant is (unfortunately) used for matrices as well as for their determinants. The matrix could be called circular instead.
The eigenvalues of a circulant n X n matrix M(n) are lambda^{(n)}_k=sum(a_j*(rho_n)^(j*k),j=1..n), with the n-th roots of unity (rho_n)^k, k=1..n, where rho_n:=exp(2*Pi/n). See the P. J. Davis reference which uses a different convention.

			n=4: the circulant 4 X 4 matrix is M(4) = matrix([3,2,1,1],[1,3,2,1],[1,1,3,2],[2,1,1,3]).
n=4: 4th roots of unity: rho_4 = I, (rho_4)^2 = -1, (rho_4)^3 = -I, (rho_4)^4 =1, with I^2=-1.
n=4: the eigenvalues of M(4) are therefore 1*I^k + 1*(-1)^k + 2*(-I)^k + 3*1^k, k=1,...,4, namely 2-I,1,2+I,7.
n=4: a(4)= Det(M(4)) = 35 = (2-I)*1*(2+I)*7.

P. J. Davis, Circulant Matrices, J. Wiley, New York, 1979.

Cf. A123744 (Fibonacci circulants including F_0 = 0).

Cf. A052182 (with n instead of Fibonacci(n) and first row reversed).

a(n) = matdet(matrix(n, n, i, j, fibonacci(n-lift(Mod(j-i, n))))); \\ Michel Marcus, Aug 11 2019

a(n) = product(lambda^{(n)}k,k=1..n), with lambda^{(n)}_k=sum(F{j}*(rho_n)^(j*k),j=1..n).

a(n) = C_n([F_{n},F_{n-2},...,F_1]) with the Fibonacci numbers F_n:=A000045(n) and the circulant C_n (see comment above).

A383772 a(n) = neg(M(n)), where M(n) is the n X n circulant matrix with (row 1) = (1, 2, ... , n), and neg(M(n)) is the negative part of the determinant of M(n); see A380661.

Original entry on oeis.org

0, -4, -18, -610, -15675, -772122, -47282844, -3918873376, -410168886615, -53329052728000, -8417451284317614, -1586200451151892608, -351735180091505203539, -90667510133054591492224, -26884188746929397888775000, -9086147134545912835276742656

Offset: 1

Clark Kimberling, May 15 2025

sign

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

Cf. A052182 (determinant), A085719 (permanent), A380661, A383773, A383774, A383775.

z = 18;
v[n_] := Table[k + 1, {k, 0, n - 1}];
u[n_] := Table[RotateRight[#, 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 *)
neg = (d - p)/2  (* A383772 *)
pos = (d + p)/2  (* A383773 *)

A383773 a(n) = pos(M(n)), where M(n) is the n X n 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, 36, 450, 17550, 744906, 47753440, 3909436192, 410384120220, 53323552728000, 8417606908865220, 1586195621597483136, 351735343178101060906, 90667504180193792086144, 26884188980472806091900000, 9086147124746080046118543360, 3472279409772212369077001352888

Offset: 1

Clark Kimberling, May 17 2025

nonn

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

Cf. A052182 (determinant), A085719 (permanent), A380661, A383772, A383774, A383775.

z = 18;
v[n_] := Table[k + 1, {k, 0, n - 1}];
u[n_] := Table[RotateRight[#, 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 *)
neg = (d - p)/2   (* A383772 *)
pos = (d + p)/2  (* A383773 *)

cp's OEIS Frontend

A048954 Wendt determinant of n-th circulant matrix C(n).

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A066933 Determinant of n X n matrix whose rows are cyclic permutations of 2..prime(n).

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A086459 Determinant of the circulant matrix whose rows are formed by successively rotating the vector (1, 2, 4, 8, ..., 2^(n-1)) right.

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A070896 Determinant of the Cayley addition table of Z_{n}.

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A118713 a(n) = determinant of n X n circulant matrix whose first row is A001358(1), A001358(2), ..., A001358(n) where A001358(n) = n-th semiprime.

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A023999 Absolute value of determinant of n X n matrix whose entries are the integers from 1 to n^2 spiraling inward, starting in a corner.

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