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 1-10 of 18 results. Next

A384076 a(n) = pos(M(n)), where M(n) is the n X n circulant matrix with (row 1) = (1,3,5,7, ..., 2n - 1), and pos(M(n)) is the positive part of the determinant of M(n); see A380661.

Original entry on oeis.org

1, 1, 153, 2668, 250200, 19423560, 2515242520, 404114856640, 84196030473216, 21703670967664000, 6808856052755927808, 2552126898198385479168, 1126590812208410998119424, 578462173661889165983466496, 341831898528862885226121600000
Offset: 1

Views

Author

Clark Kimberling, May 22 2025

Keywords

Examples

			The rows of M(4) are (1,3,5,7), (7,1,3,5), (5,7,1,3), (3,5,7,1); determinant(M(4)) = -4716; permanent(M(4)) = 2668, so neg(M(4)) = (-2048 - 7384)/2 = -4716 and pos(M(4)) = (-2048 + 7384)/2 = 2668.

Crossrefs

Cf. A193678 (determinant), A384075 (permanent), A380661, A384077, A384078.

Programs

  • Mathematica
    z = 19;
v[n_] := Table[2 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}]   (* A384074  *)
d = Table[Simplify[Det[u[n]]], {n, 1, z}]  (* A193678,, with alternating signs *)
neg = (d - p)/2   (* A384075 *)
pos = (d + p)/2   (* A384076 *)

Formula

a(n) = (1/2)*(-(-1)^n*A193678(n) + A384074(n)).

A384077 a(n) = neg(M(n)), where M(n) is the n X n left circulant matrix with (row 1) = (1,3,5,7, ..., 2n - 1), and neg(M(n)) is the negative part of the determinant of M(n); see A380661.

Original entry on oeis.org

0, -9, -153, -2668, -200200, -20916552, -2515242520, -404114856640, -84096850828032, -21708790967664000, -6808856052755927808, -2552126898198385479168, -1126589571631974396251136, -578462264691449080954733568, -341831898528862885226121600000
Offset: 1

Views

Author

Clark Kimberling, May 29 2025

Keywords

Examples

			The rows of M(4) are (1,3,5,7), (3,5,7,1), (5,7,1,3), (7,1,3,5); determinant(M(4)) = 2048; permanent(M(4)) = 7384, so neg(M(4)) = (7384 - 2048)/2 = -2668 and pos(M(4)) = (7384+2048)/2 = 4716.

Crossrefs

Cf. A193678 (determinant), A384074 (permanent), A380661, A384076, A384078.

Programs

  • Mathematica
    z = 15;
v[n_] := Table[2 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}]   (* A384074 *)
d = Table[Simplify[Det[u[n]]], {n, 1, z}] (* A193678 up to signs *)
neg = (d - p)/2    (* A384077 *)
pos = (d + p)/2    (* A384078 *)

Formula

a(n) = (1/2)*(s(n)*A193678(n) - A384074(n)), where s(n) = (-1)^((2*n+(-1)^n-1)/4).

A384078 a(n) = pos(M(n)), where M(n) is the n X n left circulant matrix with (row 1) = (1,3,5,7, ..., 2n - 1), and pos(M(n)) is the positive part of the determinant of M(n); see A380661.

Original entry on oeis.org

1, 1, 45, 4716, 250200, 19423560, 2462535768, 406262340288, 84196030473216, 21703670967664000, 6808563893605222144, 2552145158372103507456, 1126590812208410998119424, 578462173661889165983466496, 341831891354409385226121600000
Offset: 1

Views

Author

Clark Kimberling, Jun 01 2025

Keywords

Examples

			The rows of M(4) are (1,3,5,7), (3,5,7,1), (5,7,1,3), (7,1,3,5); determinant(M(4)) = 2048; permanent(M(4)) = 7384, so neg(M(4)) = (7384 - 2048)/2 = -2668 and pos(M(4)) = (7384+2048)/2 = 4716.

Crossrefs

Cf. A193678 (determinant), A384074 (permanent), A380661, A384076, A384077.

Programs

  • Mathematica
    z = 15;
v[n_] := Table[2 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}]     (* A384074 *)
d = Table[Simplify[Det[u[n]]], {n, 1, z}] (* A193678 up to signs *)
neg = (d - p)/2    (* A384077 *)
pos = (d + p)/2    (* A384078 *)

Formula

a(n) = (1/2)*(s(n)*A193678(n) - A384074(n)), where s(n) = (-1)^((2*n+(-1)^n-1)/4).

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

Views

Author

Clark Kimberling, May 15 2025

Keywords

Examples

			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.

Crossrefs

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

Programs

  • Mathematica
    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

Views

Author

Clark Kimberling, May 17 2025

Keywords

Examples

			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.

Crossrefs

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

Programs

  • Mathematica
    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 *)

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.

Original entry on oeis.org

0, -4, -36, -450, -15675, -772122, -47753440, -3909436192, -410168886615, -53329052728000, -8417606908865220, -1586195621597483136, -351735180091505203539, -90667510133054591492224, -26884188980472806091900000, -9086147124746080046118543360
Offset: 1

Views

Author

Clark Kimberling, May 17 2025

Keywords

Examples

			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.

Crossrefs

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

Programs

  • Mathematica
    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

Views

Author

Clark Kimberling, May 22 2025

Keywords

Examples

			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.

Crossrefs

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

Programs

  • Mathematica
    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 *)

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.

A384075 a(n) = neg(M(n)), where M(n) is the n X n circulant matrix with (row 1) = (1,3,5,7, ..., 2n - 1), and neg(M(n)) is the negative part of the determinant of M(n); see A380661.

Original entry on oeis.org

0, -9, -45, -4716, -200200, -20916552, -2462535768, -406262340288, -84096850828032, -21708790967664000, -6808563893605222144, -2552145158372103507456, -1126589571631974396251136, -578462264691449080954733568, -341831891354409385226121600000
Offset: 1

Views

Author

Clark Kimberling, May 22 2025

Keywords

Examples

			The rows of M(4) are (1,3,5,7), (7,1,3,5), (5,7,1,3), (3,5,7,1); determinant(M(4)) = -4716; permanent(M(4)) = 2668, so neg(M(4)) = (-2048 - 7384)/2 = -4716 and pos(M(4)) = (-2048 + 7384)/2 = 2668.

Crossrefs

Cf. A193678 (determinant), A384074 (permanent), A380661, A384076, A384077, A384078.

Programs

  • Mathematica
    z = 19;
v[n_] := Table[2 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}]   (* A384074  *)
d = Table[Simplify[Det[u[n]]], {n, 1, z}]  (* A193678, with alternating signs *)
neg = (d - p)/2   (* A384075 *)
pos = (d + p)/2   (* A384076 *)

Formula

a(n) = (1/2)*((-1)^n*A193678(n) - A384074(n)).

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.
Showing 1-10 of 18 results. Next

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