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 1-4 of 4 results.

A123744 Circulants of Fibonacci numbers (including F_0 = 0).

Original entry on oeis.org

1, 0, 1, 2, 16, 287, 16128, 2192140, 830952837, 805644641664, 2080690769701456, 14002804169885909807, 247753675148653634781184, 11469641168045182197979378136, 1391545878431673359565624090480585, 442017027765434652128920030338417270784, 367683484076057642925500106042968712221296320
Offset: 0

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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. Hence C_n=C_n([a_n,a_{n-1},...,a_1]), with the first row of M given.
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 circular 4 X 4 matrix is M(4) = matrix([[2,1,1,0],[0,2,1,1],[1,0,2,1],[1,1,0,2]]).
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. A123744 n=4: the eigenvalues of M(4) are therefore: 0*I^k + 1*(-1)^k + 1*(-I)^k + 2*1^k, k=1,...,4, namely 1-I, 2, 1+I, 4.
n=4: a(4)= Det(M(4)) = 16 = (1-I)*2*(1+I)*4.

References

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

Crossrefs

Cf. A123745 (other Fibonacci circulants without F_0 = 0).
Cf. A081131 (with n instead of Fibonacci(n)).
Cf. A000045.

Programs

  • PARI
    mm(n) = matdet(matrix(n, n, i, j, fibonacci(n-1-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-1}*(rho_n)^(j*k),j=1..n).
a(n) = C_n([F_{n-1},F_{n-2},...,F_0]) with the Fibonacci numbers F_n:=A000045(n) and the circulant C_n (see comment above).

Extensions

More terms from Michel Marcus, Aug 11 2019
a(0)=1 prepended by Alois P. Heinz, Jun 27 2025

A384592 a(n) = neg(M(n)), where M(n) is the n X n circulant matrix with (row 1) = (F(1), F(2), ..., F(n)), 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, -6, -150, -5370, -800640, -190224111, -176905217601, -291774510700416, -1732081174959242854, -20169113574187183899414, -780617739418078727245260288, -64339950490716309261029142530799, -16501635484912814610105139633118406681
Offset: 1

Views

Author

Clark Kimberling, Jul 02 2025

Keywords

Examples

			The rows of M(4) are (1,1,2,3), (3,1,1,2), (23,1,1), (1,2,3,1); determinant(M(4)) = -35; permanent(M(4)) = 265, so neg(M(4)) = (-35 -265)/2 = -150 and pos(M(4)) = (-35 + 265)/2 = 115.

Crossrefs

Cf. A000045, A380661, A123745 (determinant), A384591 (permanent), A384593.

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}]   (* A384591  *)
d = Table[Simplify[Det[u[n]]], {n, 1, z}] (* A123745, with alternating signs *)
neg = (d - p)/2     (* A384592 *)
pos = (d + p)/2     (* A384593 *)

Formula

a(n) = (1/2)*((-1)^n A123745(n) - A384591(n)).

A384593 a(n) = pos(M(n)), where M(n) is the n X n circulant matrix with (row 1) = (F(1), F(2), ..., F(n)), 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

1, 1, 10, 115, 7182, 630400, 236525784, 146492200737, 343945864714624, 1504008427530969535, 22752527891269251753118, 704885020930148344661403136, 70113811460119152088048405685808, 15355281759224121782879344275585373609
Offset: 1

Views

Author

Clark Kimberling, Jul 02 2025

Keywords

Examples

			The rows of M(4) are (1,1,2,3), (3,1,1,2), (2,3,1,1), (1,2,3,1); determinant(M(4)) = -35; permanent(M(4)) = 265, so neg(M(4)) = (-35 - 265)/2 = -150 and pos(M(4)) = (-35 + 265)/2 = 115.

Crossrefs

Cf. A000045, A380661, A123745 (determinant), A384591 (permanent), A384592.

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}]   (* A384591  *)
d = Table[Simplify[Det[u[n]]], {n, 1, z}] (* A123745, with alternating signs *)
neg = (d - p)/2     (* A384592 *)
pos = (d + p)/2     (* A384593 *)

Formula

a(n) = (1/2)*((-1)^n A123745(n) + A384591(n)).

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 1-4 of 4 results.

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