cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

Views

Author

Wolfdieter Lang, Nov 10 2006, Jan 27 2009

Keywords

Comments

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.

Examples

			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.

References

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

Crossrefs

Cf. A123744 (Fibonacci circulants including F_0 = 0).
Cf. A052182 (with n instead of Fibonacci(n) and first row reversed).

Programs

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

Formula

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).

A384080 a(n) = neg(M(n)), where M(n) is the n X n circulant matrix with (row 1) = (F(0), F(1), ..., F(n-1)), where F = A000045 (Fibonacci numbers), and neg(M(n)) is the negative part of the determinant of M(n); see A380661.

Original entry on oeis.org

0, -1, 0, -25, -295, -43264, -5469632, -3628008315, -3569061677472, -13761972434293885, -98350155131379362607, -2395228216526569309464064, -121960521137098218596500559704, -19460957348767631231695727354978359, -6994735829985160817748505807288716492800
Offset: 1

Views

Author

Clark Kimberling, Jun 19 2025

Keywords

Examples

			The rows of M(4) are (0,1,1,2), (2,0,1,1), (1,2,0,1), (1,1,2,0); determinant(M(4)) = -16; permanent(M(4)) = 34, so neg(M(4)) = (-16 - 34)/2 = -25 and pos(M(4)) = (-16 + 34)/2 = 9.

Crossrefs

Cf. A123744 (determinant), A384079 (permanent), A380661, A384313.

Programs

  • Mathematica
    z = 14;
v[n_] := Table[Fibonacci[k], {k, 0, n - 1}];
u[n_] := Table[RotateRight[#, k - 1], {k, 1, Length[#]}] &[v[n]]
p = Table[Permanent[u[n]], {n, 1, z}]   (* A384079 *)
d = Table[Simplify[Det[u[n]]], {n, 1, z}] (* A123744, with alternating signs *)
neg = (d - p)/2    (* A384080 *)
pos = (d + p)/2    (* A384313 *)

Formula

a(n) = (-(-1)^n * A123744(n) - A384079(n)) / 2.

A384079 a(n) = permanent of the n X n circulant matrix with (row 1) = (F(0), F(1), ..., F(n-1)), where F = A000045 (Fibonacci numbers).

Original entry on oeis.org

1, 0, 1, 2, 34, 877, 70400, 13131404, 6425063793, 7943767996608, 25443254098886314, 210703114432644635021, 4542702757904484984146944, 255390683442241619390980497544, 37530368819103589103825830619476133, 14431488687735756287625931644915850256384
Offset: 0

Views

Author

Clark Kimberling, Jun 01 2025

Keywords

Crossrefs

Cf. A000045, A123744 (determinant), A384080.

Programs

  • Mathematica
    z = 16;
v[n_] := Table[Fibonacci[k], {k, 0, n - 1}];
u[n_] := Table[RotateRight[#, k - 1], {k, 1, Length[#]}] &[v[n]]
Table[Simplify[Permanent[u[n]]], {n, 1, z}]

Extensions

a(0)=1 prepended by Alois P. Heinz, Jun 27 2025

A384313 a(n) = pos(M(n)), where M(n) is the n X n circulant matrix with (row 1) = (F(0), F(1), ..., F(n-1)), where F = A000045 (Fibonacci numbers), and pos(M(n)) is the positive part of the determinant of M(n); see A380661.

Original entry on oeis.org

0, 0, 2, 9, 582, 27136, 7661772, 2797055478, 4374706319136, 11681281664592429, 112352959301265272414, 2147474541377915674682880, 133430162305143400794479937840, 18069411470335957872130103264497774, 7436752857750595469877425837627133763584
Offset: 1

Views

Author

Clark Kimberling, Jun 27 2025

Keywords

Examples

			The rows of M(4) are (0,1,1,2), (2,0,1,1), (1,2,0,1), (1,1,2,0); determinant(M(4)) = -16; permanent(M(4)) = 34, so neg(M(4)) = (-16 - 34)/2 = -25 and pos(M(4)) = (-16 + 34)/2 = 9.

Crossrefs

Cf. A123744 (determinant), A384079 (permanent), A380661, A384080.

Programs

  • Mathematica
    z = 14;
v[n_] := Table[Fibonacci[k], {k, 0, n - 1}];
u[n_] := Table[RotateRight[#, k - 1], {k, 1, Length[#]}] &[v[n]]
p = Table[Permanent[u[n]], {n, 1, z}]   (* A384079 *)
d = Table[Simplify[Det[u[n]]], {n, 1, z}] (* A123744 *)
neg = (d - p)/2    (* A384080 *)
pos = (d + p)/2    (* A384313 *)

Formula

a(n) = (-(-1)^n * A123744(n) + A384079(n)) / 2.

A384591 a(n) = permanent of the n X n circulant matrix with (row 1) = (F(1), F(2), ..., F(n)), where F = A000045 (Fibonacci numbers).

Original entry on oeis.org

1, 1, 2, 16, 265, 12552, 1431040, 426749895, 323397418338, 635720375415040, 3236089602490212389, 42921641465456435652532, 1485502760348227071906663424, 134453761950835461349077548216607, 31856917244136936392984483908703780290
Offset: 0

Views

Author

Clark Kimberling, Jun 27 2025

Keywords

Crossrefs

Cf. A000045, A123744 (determinant), A384080, A384313.

Programs

  • Mathematica
    z = 14;
v[n_] := Table[Fibonacci[k], {k, 1, n}];
u[n_] := Table[RotateRight[#, k - 1], {k, 1, Length[#]}] &[v[n]]
Table[Permanent[u[n]], {n, 1, z}]

A145570 Circulants of Catalan numbers.

Original entry on oeis.org

1, 0, 4, 459, 474743, 5237087765, 686666209113536, 1140713637429903585344, 24957776794187383667855422048, 7377122100200717681983830999516060000, 30004208141654594144715773978429859682880072414, 1703184026083327296951313841743251806796128938200000000000, 1365027457901516492029047382022588117973123824294791438142988114734512
Offset: 1

Views

Author

Wolfdieter Lang Feb 05 2009

Keywords

Comments

A circulant C_n is the determinant of a circular n X n matrix M, i.e. one with entries M_{i,j}=a_{i-j} where the indices are taken mod n. Hence C_n=C_n([a_n=a_0,a_{n-1},...,a_1]), with the first row of M given. The second row is [a_1,a_0,a_{n-1},...,a_2], etc.
The eigenvalues of a circular 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, and the term circulant for circular.

Examples

			n=4: the circular 4 X 4 matrix is M(4) = matrix([[5,2,1,1],[1,5,2,1],[1,1,5,2],[2,1,1,5]]).
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 + 5*1^k, k=1,..,4, namely 4-I, 3, 4+I, 9.
n=4: a(4)= Det(M(4)) = (4-I)*3*(4+I)*9 = 459.

References

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

Crossrefs

A123744, A123745 (circulants for Fibonacci numbers).

Programs

  • Mathematica
    rho[n_] := Exp[2*I*Pi/n]; lambda[n_, k_] := Sum[ CatalanNumber[j - 1]*rho[n]^(j*k), {j, 1, n}]; a[n_?EvenQ] := FullSimplify[ Product[ lambda[n, k], {k, 1, n}]]; a[n_?OddQ] := Expand[ Product[ lambda[n, k], {k, 1, n}]] /. Plus[x_Integer, Times[y_Integer, Power[E, Times[ Complex[0, Rational[, FactorInteger[n][[1, 1]]]], Pi]]], _] -> x - y; Table[a[n], {n, 1, 13}] (* Jean-François Alcover, Sep 27 2011 *)

Formula

a(n)=product(lambda^{(n)}k,k=1..n), with lambda^{(n)}_k=sum(Ca{j-1}*(rho_n)^(j*k), j=1..n).
a(n) = C_n([Ca_{n-1},Ca_{n-2},...,Ca_0]) with the Catalan numbers Ca_n:=A000108(n), and the circulant C_n (see comment above).
Showing 1-6 of 6 results.

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