A048954 -id:A048954 - cp's OEIS frontend

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).

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

A096964 Wendt's determinant of n.

Original entry on oeis.org

-3, -3, 0, -375, -3993, 0, -15265344, -1343091375, 0, -210736858987743, -141498026224804329, 0, -987345386156157037417593, -4875797582053878382039400448, 0, -1562716604740038367719196682456673375

Offset: 1

Ralf Stephan, Aug 01 2004

sign

a(n) = 0 for multiples of 3.

See also A048954 for a different definition.

D. Ford and V. Jha, On Wendt's determinant and Sophie Germain's Theorem, Experiment. Math., Volume 2, Issue 2 (1993), 113-120.

PARI
```
a(n)=polresultant(x^n-1,(-1-x)^n-1,x)
```

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

A215656 Solution R of (2u)^2 = R^2 - pS^2, where p is the n-th prime of the form 4k+1.

Original entry on oeis.org

147, 20522387091091, 89544370675021535714607142, 8801866915656397716021519532258687362772409962179980790374047406788427

Offset: 1

Jonathan Sondow, Aug 19 2012

nonn

p = A002144(n), u = A215615(p), and S = A215657(n).
A215615 is computed from Wendt's circulant determinant A048954.
Brown and Chamberland (2012, p. 600) give explicit formulas for u, R, S.

			2*A215615(5) = 2*11 = 22 and 22^2  = 147^2 - 5*65^2, so a(1) = 147.

Ezra Brown and Marc Chamberland, Generalizing Gauss's gem, Amer. Math. Monthly, 119 (Aug. 2012), 597-601.

Cf. A002144, A048954, A215615, A215657.

a(n) = sqrt(4*u^2 + p*S^2) with S = A215657(n), p = A002144(n), u = A215615(p).

A215657 Solution S of (2u)^2 = R^2 - pS^2, where p is the n-th prime of the form 4k+1.

Original entry on oeis.org

65, 5691884464123, 2171769991015128035203320, 1634465653492219202324217583600006782459921190308836446038375668451525

Offset: 1

Jonathan Sondow, Aug 20 2012

nonn

p = A002144(n), u = A215615(p), and R = A215656(n).
A215615 is computed from Wendt's circulant determinant A048954.
Brown and Chamberland (2012, p. 600) give explicit formulas for u, R, S.

			2*A215615(5) = 2*11 = 22 and 22^2  = 147^2 - 5*65^2, so a(1) = 65.

Ezra Brown and Marc Chamberland, Generalizing Gauss's gem, Amer. Math. Monthly, 119 (Aug. 2012), 597-601.

Cf. A002144, A048954, A215615, A215656.

a(n) = sqrt((R^2 - 4*u^2)/p) with R = A215656(n), p = A002144(n), u = A215615(p).

A118702 a(n) = determinant of n X n circulant matrix whose first row is the first n Lucas numbers A000032, from L(0) to L(n-1).

Original entry on oeis.org

2, 3, 18, 0, 8347, -861952, 391524998, -359089453125, 893329160995712, -5499366235206395112, 87687141416511254851323, -3590079701896396800000000000, 381284797406693371431803926245802, -105147887074796935457211770823970013737

Offset: 1

Jonathan Vos Post, May 20 2006

			a(4) = 0 because of the singular matrix:
[2, 1, 3, 4]
[4, 2, 1, 3]
[3, 4, 2, 1]
[1, 3, 4, 2].

Eric Weisstein's World of Mathematics, Circulant Matrix.

A000032 Lucas numbers (beginning at 2): L(n) = L(n-1) + L(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. A000032, A048954, A052182, A066933, A086459, A086569.

circ[w_] := NestList[RotateRight, w, Length[w] - 1]; Table[ Det[ circ[ LucasL@ Range[0, n - 1]]], {n, 10}] (* Giovanni Resta, Jun 16 2016 *)

Corrected and extended by Giovanni Resta, Jun 16 2016

cp's OEIS Frontend

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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A096964 Wendt's determinant of n.

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PARI

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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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.

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A215656 Solution R of (2*u)^2 = R^2 - p*S^2, where p is the n-th prime of the form 4k+1.

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A215657 Solution S of (2*u)^2 = R^2 - p*S^2, where p is the n-th prime of the form 4k+1.

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A215656 Solution R of (2u)^2 = R^2 - pS^2, where p is the n-th prime of the form 4k+1.

A215657 Solution S of (2u)^2 = R^2 - pS^2, where p is the n-th prime of the form 4k+1.