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 1-8 of 8 results.

A333706 Number T(n,k) of permutations p of [n] such that |p(i+k) - p(i)| <> k for i in [n-k]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 4, 6, 0, 2, 16, 20, 24, 0, 14, 44, 80, 108, 120, 0, 90, 200, 384, 544, 672, 720, 0, 646, 1288, 2240, 3264, 4128, 4800, 5040, 0, 5242, 9512, 15424, 23040, 28992, 34752, 38880, 40320, 0, 47622, 78652, 123456, 176832, 231936, 280512, 323520, 352800, 362880
Offset: 0

Views

Author

Alois P. Heinz, Apr 02 2020

Keywords

Comments

T(n,k) is defined for n,k >= 0. The triangle contains only the terms with k<=n. T(n,k) = n! for k>=n.

Examples

			Triangle T(n,k) begins:
  1;
  0,    1;
  0,    0,    2;
  0,    0,    4,     6;
  0,    2,   16,    20,    24;
  0,   14,   44,    80,   108,   120;
  0,   90,  200,   384,   544,   672,   720;
  0,  646, 1288,  2240,  3264,  4128,  4800,  5040;
  0, 5242, 9512, 15424, 23040, 28992, 34752, 38880, 40320;
  ...

Links

Crossrefs

Columns k=0-10 (for n>=k) give: A000007, A002464, A110128, A117574, A189255, A189256, A189271, A360384, A360386, A360462, A360463.
Main diagonal gives A000142.
T(2n,n) gives A189849.
T(n+1,n) gives 4*A138772(n).
T(n+2,n) gives 16*A333804(n).
Cf. A000170 (condition is satisfied for all k), A248686 (p(i) at distance k are sorted).

A137774 Number of ways to place n nonattacking empresses on an n X n board.

Original entry on oeis.org

1, 2, 2, 8, 20, 94, 438, 2766, 19480, 163058, 1546726, 16598282, 197708058, 2586423174, 36769177348, 563504645310, 9248221393974, 161670971937362, 2996936692836754, 58689061747521430, 1210222434323163704, 26204614054454840842, 594313769819021397534, 14086979362268860896282
Offset: 1

Views

Author

Vaclav Kotesovec, Jan 27 2011

Keywords

Comments

An empress moves like a rook and a knight.

Links

Crossrefs

Cf. A201513, A000170, A002465, A201540.
Cf. A185085, A051223, A244284, A201511, A201861, A137774, A245011.
Cf. A218244, A002464, A110128, A117574, A089222, A002493.

Formula

Asymptotics (Vaclav Kotesovec, Jan 26 2011): a(n)/n! -> 1/e^4.
General asymptotic formulas for number of ways to place n nonattacking pieces rook + leaper[r,s] on an n X n board:
a(n)/n! -> 1/e^2 for 0
a(n)/n! -> 1/e^4 for 0

Extensions

Terms a(16)-a(17) from Vaclav Kotesovec, Feb 06 2011
Terms a(18)-a(19) from Wolfram Schubert, Jul 24 2011
Terms a(20)-a(24) (computed by Wolfram Schubert), Vaclav Kotesovec, Aug 25 2012

A110128 Number of permutations p of 1,2,...,n satisfying |p(i+2)-p(i)| not equal to 2 for all 0

Original entry on oeis.org

1, 1, 2, 4, 16, 44, 200, 1288, 9512, 78652, 744360, 7867148, 91310696, 1154292796, 15784573160, 232050062524, 3648471927912, 61080818510972, 1084657970877416, 20361216987032284, 402839381030339816, 8377409956454452732
Offset: 0

Views

Author

Roberto Tauraso, A. Nicolosi and G. Minenkov, Jul 13 2005

Keywords

Comments

When n is even: 1) Number of ways that n persons seated at a rectangular table with n/2 seats along the two opposite sides can be rearranged in such a way that neighbors are no more neighbors after the rearrangement. 2) Number of ways to arrange n kings on an n X n board, with 1 in each row and column, which are non-attacking with respect to the main four quadrants.
a(n) is also number of ways to place n nonattacking pieces rook + alfil on an n X n chessboard (Alfil is a leaper [2,2]) [From Vaclav Kotesovec, Jun 16 2010]
Note that the conjectured recurrence was based on the 600-term b-file, not the other way round. - N. J. A. Sloane, Dec 07 2022

Links

Crossrefs

Cf. A089222, A002464, A117574, A189281.
Column k=2 of A333706.

Formula

A formula is given in the Tauraso reference.
Asymptotic (R. Tauraso 2006, quadratic term V. Kotesovec 2011): a(n)/n! ~ (1 + 4/n + 8/n^2)/e^2.
a(n) ~ exp(-2) * n! * (1 + 4/n + 8/n^2 + 68/(3*n^3) + 242/(3*n^4) + 1692/(5*n^5) + 72802/(45*n^6) + 2725708/(315*n^7) + 16083826/(315*n^8) + 186091480/(567*n^9) + 32213578294/(14175*n^10) + ...), based on the recurrence by Manuel Kauers. - Vaclav Kotesovec, Dec 05 2022

Extensions

Edited by N. J. A. Sloane at the suggestion of Vladeta Jovovic, Jan 01 2008
Terms a(33)-a(35) from Vaclav Kotesovec, Apr 20 2012

A189255 Number of permutations p of 1,2,...,n satisfying |p(i+4)-p(i)|<>4 for all 1<=i<=n-4.

Original entry on oeis.org

1, 2, 6, 24, 108, 544, 3264, 23040, 176832, 1563392, 15536160, 171172224, 2066033472, 27146652480, 385447394880, 5878028516736, 95776238793504, 1660164417866304, 30496085473606944, 591661117634375040, 12087628978334638752
Offset: 1

Views

Author

Vaclav Kotesovec, Apr 19 2011

Keywords

Comments

a(n) is also number of ways to place n nonattacking pieces rook + leaper[4,4] on an n X n chessboard.

Links

Crossrefs

Cf. A002464, A110128, A117574.
Column k=4 of A333706.

Formula

Asymptotic (R. Tauraso 2006, quadratic term V. Kotesovec 2011): a(n)/n! ~ (1 + 12/n + 64/n^2)/e^2.

Extensions

Terms a(26)-a(27) from Vaclav Kotesovec, Apr 20 2012

A189256 Number of permutations p of 1,2,...,n satisfying |p(i+5)-p(i)|<>5 for all 1<=i<=n-5.

Original entry on oeis.org

1, 2, 6, 24, 120, 672, 4128, 28992, 231936, 2088960, 20434944, 221871360, 2645370624, 34344038400, 482103767040, 7269498483456, 117240911729664, 2013265377314688, 36665783917283328, 705762463906133760, 14313891805008665856
Offset: 1

Views

Author

Vaclav Kotesovec, Apr 19 2011

Keywords

Comments

a(n) is also number of ways to place n nonattacking pieces rook + leaper[5,5] on an n X n chessboard.

Links

Crossrefs

Cf. A002464, A110128, A117574, A189255.
Column k=5 of A333706.

Formula

Asymptotic (R. Tauraso 2006, quadratic term V. Kotesovec 2011): a(n)/n! ~ (1 + 16/n + 110/n^2)/e^2.

Extensions

Terms a(25)-a(26) from Vaclav Kotesovec, Apr 20 2012

A189282 Number of permutations p of 1,2,...,n satisfying p(i+3)-p(i)<>3 for all 1<=i<=n-3.

Original entry on oeis.org

1, 1, 2, 6, 22, 98, 534, 3414, 25498, 217338, 2080990, 22076030, 256888218, 3252308706, 44497313158, 654139144158, 10281397705242, 172033123244330, 3052895403376110, 57266799403366334, 1132124282036449570, 23524895818926592242
Offset: 0

Views

Author

Vaclav Kotesovec, Apr 19 2011

Keywords

Comments

a(n) is also number of ways to place n nonattacking pieces rook + semi-leaper[3,3] on an n X n chessboard.

Links

Crossrefs

Cf. A000255, A189281, A117574.

Formula

Asymptotics: a(n)/n! ~ (1 + 5/n + 6/n^2)/e.

A189271 Number of permutations p of 1,2,...,n satisfying |p(i+6)-p(i)|<>6 for all 1<=i<=n-6.

Original entry on oeis.org

1, 2, 6, 24, 120, 720, 4800, 34752, 280512, 2528256, 25282560, 278323200, 3289036800, 42336448512, 589351062528, 8820501301248, 141215147788800, 2407845089203200, 43543159894318080, 832618225074748416, 16782891792284791296
Offset: 1

Views

Author

Vaclav Kotesovec, Apr 19 2011

Keywords

Comments

a(n) is also number of ways to place n nonattacking pieces rook + leaper[6,6] on an n X n chessboard.

Links

Crossrefs

Cf. A002464, A110128, A117574, A189255, A189256.
Column k=6 of A333706.

Formula

Asymptotic (R. Tauraso 2006, quadratic term V. Kotesovec 2011): a(n)/n! ~ (1 + 20/n + 168/n^2)/e^2.
Generally (for this sequence is d=6): 1/e^2*(1+4(d-1)/n+2d*(3d-4)/n^2+...).

Extensions

Terms a(23)-a(24) from Vaclav Kotesovec, Apr 21 2012

A189839 Number of ways to place n nonattacking composite pieces rook + rider[3,3] on an n X n chessboard.

Original entry on oeis.org

1, 2, 6, 20, 80, 384, 2112, 12992, 94272, 716800, 6141440, 58451568, 596647568, 6555879072, 77766001056, 981202169600
Offset: 1

Views

Author

Vaclav Kotesovec, Apr 29 2011

Keywords

Comments

a(n) is also number of permutations p of 1,2,...,n satisfying |p(j+3k)-p(j)|<>3k for all j>=1, k>=1, j+3k<=n

Links

Crossrefs

Cf. A117574, A000170, A189838
Showing 1-8 of 8 results.

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

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