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 1-5 of 5 results.

A212796 Square array read by antidiagonals: T(m,n) = number of spanning trees in C_m X C_n.

Original entry on oeis.org

1, 2, 2, 3, 32, 3, 4, 294, 294, 4, 5, 2304, 11664, 2304, 5, 6, 16810, 367500, 367500, 16810, 6, 7, 117600, 10609215, 42467328, 10609215, 117600, 7, 8, 799694, 292626432, 4381392020, 4381392020, 292626432, 799694, 8, 9, 5326848, 7839321861, 428652000000, 1562500000000, 428652000000, 7839321861, 5326848, 9
Offset: 1

Views

Author

N. J. A. Sloane, May 27 2012

Keywords

Examples

			Array begins:
  1,    2,      3,        4,          5,            6               7, ...
  2,   32,    294,     2304,      16810,       117600,         799694, ...
  3,  294,  11664,   367500,   10609215,    292626432,     7839321861, ...
  4, 2304, 367500, 42467328, 4381392020, 428652000000, 40643137651228, ...
  ...

Links

Crossrefs

Rows and columns 1..10 give A000027, A212797, A212798, A212799, A358810, A358811, A358812, A358813, A358814, A358815.
Diagonal gives A212800.
Cf. A116469, A173958, A340560.

Programs

  • Maple
    Digits:=200;
T:=(m,n)->round(Re(evalf(simplify(expand(
m*n*mul(mul( 4*sin(h*Pi/m)^2+4*sin(k*Pi/n)^2, h=1..m-1), k=1..n-1))))));
  • PARI
    default(realprecision, 120);
{T(n, k) = round(n*k*prod(a=1, n-1, prod(b=1, k-1, 4*sin(a*Pi/n)^2+4*sin(b*Pi/k)^2)))} \\ Seiichi Manyama, Jan 13 2021

Formula

T(m,n) = m*n*Prod(Prod( 4*sin(h*Pi/m)^2+4*sin(k*Pi/n)^2, h=1..m-1), k=1..n-1).

A340562 a(n) = Product_{1<=j,k<=n-1} (4*sin(j*Pi/n)^2 + 4*sin(k*Pi/n)^2).

Original entry on oeis.org

1, 8, 1296, 2654208, 62500000000, 16314248724480000, 46246966018211028668416, 1405124434459231021756179283968, 453518708737693704370173592484540645376, 1545285638496177620571506637671497728000000000000
Offset: 1

Views

Author

Seiichi Manyama, Jan 11 2021

Keywords

Crossrefs

Main diagonal of A340560.
Cf. A212800.

Programs

  • Mathematica
    Table[Product[4*Sin[j*Pi/n]^2 + 4*Sin[k*Pi/n]^2, {k, 1, n-1}, {j, 1, n-1}], {n, 1, 12}] // Round (* Vaclav Kotesovec, Feb 14 2021 *)
  • PARI
    default(realprecision, 120);
{a(n) = round(prod(j=1, n-1, prod(k=1, n-1, 4*sin(j*Pi/n)^2+4*sin(k*Pi/n)^2)))}

Formula

a(n) = A212800(n)/n^2.
a(n) ~ Gamma(1/4)^4 * exp(4*G*n^2/Pi) / (16 * Pi^3 * n^2), where G is Catalan's constant A006752. - Vaclav Kotesovec, Feb 14 2021

A335586 Number of domino tilings of a 2n X 2n toroidal grid.

Original entry on oeis.org

1, 8, 272, 90176, 311853312, 11203604497408, 4161957566985310208, 15954943354032349049274368, 630665326543010382995142219988992, 256955886436135671144699761794930161483776
Offset: 0

Views

Author

Drake Thomas, Jan 26 2021

Keywords

Comments

For n > 1, number of perfect matchings of the graph C_2n X C_2n.

Examples

			For n = 1, there are a(1) = 8 tilings (see the Links section for a diagram).

Links

Crossrefs

Number of perfect matchings of the graph C_2m X C_n: A162484 (m=1), A220864 (m=2), A232804 (m=3), A253678 (m=4), A281679 (m=5), A309018 (m=6).
Cf. A004003, A212800, A340562, A341478, A341479.

Programs

  • PARI
    default(realprecision, 120);
b(n) = round(prod(j=1, n-1, prod(k=1, n, 4*sin(j*Pi/n)^2+4*sin((2*k-1)*Pi/(2*n))^2)));
c(n) = round(prod(j=1, n, prod(k=1, n, 4*sin((2*j-1)*Pi/(2*n))^2+4*sin((2*k-1)*Pi/(2*n))^2)));
a(n) = if(n==0, 1, 4*b(n)+c(n)/2); \\ Seiichi Manyama, Feb 13 2021

Formula

a(n) = 4 * Product_{j=1..n-1} Product_{k=1..n} (4*sin(j*Pi/n)^2 + 4*sin((2*k-1)*Pi/(2*n))^2) + 1/2 * Product_{1<=j,k<=n} (4*sin((2*j-1)*Pi/(2*n))^2 + 4*sin((2*k-1)*Pi/(2*n))^2) = 4 * A341478(n)^2 + A341479(n)/2 for n > 0. - Seiichi Manyama, Feb 13 2021
a(n) ~ (1 + sqrt(2)) * exp(4*G*n^2/Pi), where G is Catalan's constant A006752. - Vaclav Kotesovec, Feb 14 2021

Extensions

More terms from Seiichi Manyama, Feb 13 2021

A229728 Decimal expansion of the square of the constant A130834.

Original entry on oeis.org

3, 2, 0, 9, 9, 1, 2, 3, 0, 0, 7, 2, 8, 1, 5, 7, 6, 7, 8, 6, 2, 9, 7, 4, 9, 4, 8, 1, 7, 7, 9, 9, 0, 5, 1, 5, 8, 7, 4, 8, 5, 9, 2, 1, 2, 4, 2, 5, 1, 8, 3, 4, 4, 9, 4, 8, 7, 4, 5, 8, 6, 0, 0, 5, 8, 4, 6, 1, 0, 2, 4, 6, 4, 1, 6, 2, 4, 2, 4, 0, 2, 0, 4, 0, 6, 6, 7, 6, 7, 1, 2, 1, 5, 1, 4, 1, 0, 8, 8, 7, 0, 9, 4, 2, 8, 4, 6, 6, 9, 1, 5, 8, 3, 8, 7, 5, 2, 2, 6, 9
Offset: 1

Views

Author

N. J. A. Sloane, Oct 01 2013

Keywords

Examples

			3.209912300728157678629749481779905158748592124251834494874586...

References

  • Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 232.

Crossrefs

Cf. A000796, A006752, A097469, A130834.
Cf. A007341, A212800, A334088, A335586, A340139, A340176, A340557, A340562, A341479.

Programs

Formula

From Amiram Eldar, Jun 12 2023: (Start)
Equals exp(4*G/Pi) = exp(4*A006752/A000796).
Equals A097469^4. (End)

A066543 Number of spanning trees in the line graph of the product of two cycle graphs, each of order n, L(C_n x C_n).

Original entry on oeis.org

782757789696, 5976745079881894723584, 29514790517935282585600000000000000, 95296975201657487970461602120230307486331043840000, 202142993853936783750487849288950496428731602354031286611374533246976
Offset: 3

Views

Author

Roberto E. Martinez II, Jan 07 2002

Keywords

Examples

			NumberOfSpanningTrees(L(C_3 x C_3)) = 782757789696

Crossrefs

Cf. A212800.

Programs

  • Mathematica
    NumberOfSpanningTrees[LineGraph[GraphProduct[Cycle[n], Cycle[n]]]] (* First load package DiscreteMath`Combinatorica` *)

Formula

a(n) = 2^(3*n^2-1) * A212800(n). - Sean A. Irvine, Oct 25 2023

Extensions

Edited by Dean Hickerson, Jan 14 2002
a(7) from Sean A. Irvine, Oct 25 2023
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.