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 11 results. Next

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

A117574 Total number of permutations p of [n] such that |p(i+3) - p(i)| is not equal to 3 for 1 <= i <= n-3.

Original entry on oeis.org

1, 1, 2, 6, 20, 80, 384, 2240, 15424, 123456, 1110928, 11287232, 127016304, 1565107248, 20935873872, 301974271248, 4669727780624, 77046043259824, 1350585114106416, 25062108668100208, 490725684463001488, 10109820295907492304
Offset: 0

Views

Author

James Sellers, Apr 27 2006

Keywords

Comments

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

Links

Crossrefs

Cf. A002464, A110128.
Column k=3 of A333706.

Formula

Formula given in Tauraso reference.
Asymptotic (R. Tauraso 2006, quadratic term V. Kotesovec 2011): a(n)/n! ~ (1 + 8/n + 30/n^2)/e^2.

Extensions

Terms a(17)-a(28) from Vaclav Kotesovec, Apr 19 2011
Terms a(29)-a(30) from Vaclav Kotesovec, Apr 20 2012
a(0)=1 prepended by Alois P. Heinz, Feb 05 2023

A189281 Number of permutations p of 1,2,...,n satisfying p(i+2) - p(i) <> 2 for all 1 <= i <= n-2.

Original entry on oeis.org

1, 1, 2, 5, 18, 75, 410, 2729, 20906, 181499, 1763490, 18943701, 222822578, 2847624899, 39282739034, 581701775369, 9202313110506, 154873904848803, 2762800622799362, 52071171437696453, 1033855049655584786, 21567640717569135515
Offset: 0

Views

Author

Vaclav Kotesovec, Apr 19 2011

Keywords

Comments

a(n) is also the number of ways to place n nonattacking pieces rook + semi-leaper (2,2) on an n X n chessboard.
Comments from Vaclav Kotesovec, Mar 05 2022: (Start)
The original submission had keyword hard because of the following running times (in 2012):
a(33) 39 hours
a(34) 78 hours
a(35) 147 hours
The conjectured recurrence would imply the asymptotic expansion for a(n)/n! ~
(1 + 3/n + 2/n^2 + 1/n^3 + 0/n^4 + 3/n^5 + 26/n^6 + 101/n^7 + 124/n^8 - 1409/n^9 - 13266/n^10)/e.
This exactly matches the formula from 2011. In addition, all coefficients are integers. It is highly probable that recurrence is correct.
(End)
There are good reasons to believe the conjecture is correct. (It has the expected form.) The problem is one of counting Hamiltonian cycles in the complement of some simple graph. There is a method for counting these efficiently (although I have not implemented in code). Similar to A242522 / A229430. - Andrew Howroyd, Mar 06 2022
See also Manuel Kauers's comments below. Since the four new terms took weeks of computation, the keyword "hard" continues to be justified. - N. J. A. Sloane, Mar 06 2022
a(40)-a(300) were computed using an independent solution (dynamic programming, O(N^4) per term), and the conjectured recurrence was further confirmed to be correct up to n=300. Consequently, the keyword "hard" is removed. - Rintaro Matsuo, Oct 18 2022

Links

Crossrefs

Cf. A000255, A002464, A055790, A110128.

Formula

Asymptotics: a(n)/n! ~ (1 + 3/n + 2/n^2)/e.
Conjectured recurrence of degree 11 and order 8: (262711*n + 1387742*n^2 - 824875*n^3 - 1855253*n^4 - 111530*n^5 + 680983*n^6 + 364242*n^7 + 84992*n^8 + 10332*n^9 + 640*n^10 + 16*n^11)*a(n) + (-1050844*n - 9705192*n^2 - 7414683*n^3 + 3536494*n^4 + 6459004*n^5 + 3326393*n^6 + 903534*n^7 + 144684*n^8 + 13756*n^9 + 720*n^10 + 16*n^11)*a(n+1) + (3492344 - 2212342*n - 8507169*n^2 - 11544227*n^3 - 12034116*n^4 - 8216995*n^5 - 3442049*n^6 - 890050*n^7 - 142300*n^8 - 13660*n^9 - 720*n^10 - 16*n^11)*a(n+2) + (19817984 + 45323852*n + 825228*n^2 - 57004661*n^3 - 57059306*n^4 - 28077270*n^5 - 8398637*n^6 - 1631510*n^7 - 207980*n^8 - 16828*n^9 - 784*n^10 - 16*n^11)*a(n+3) + (9586160 + 6680237*n - 13772613*n^2 - 27689586*n^3 - 22162455*n^4 - 9855085*n^5 - 2629562*n^6 - 427656*n^7 - 41332*n^8 - 2176*n^9 - 48*n^10)*a(n+4) + (22192864 + 44710768*n - 2924668*n^2 - 52385912*n^3 - 45161616*n^4 - 18784740*n^5 - 4549208*n^6 - 674256*n^7 - 60400*n^8 - 3008*n^9 - 64*n^10)*a(n+5) + (557152 - 2032472*n - 2937392*n^2 - 1594200*n^3 - 517688*n^4 - 122032*n^5 - 19856*n^6 - 1792*n^7 - 64*n^8)*a(n+6) + (3786960 + 7105324*n - 1191064*n^2 - 8059160*n^3 - 5938996*n^4 - 2073752*n^5 - 402736*n^6 - 44528*n^7 - 2624*n^8 - 64*n^9)*a(n+7) + (-598208 - 943004*n + 414196*n^2 + 1213772*n^3 + 728648*n^4 + 203584*n^5 + 29616*n^6 + 2176*n^7 + 64*n^8)*a(n+8) = 0. This recurrence correctly predicted the four new terms in the b-file. - Christoph Koutschan, Feb 19 2022
Comment from N. J. A. Sloane, Mar 12 2022: (Start)
The preceding conjectured recurrence is equivalent to the following, which has degree 3 and order 13, and was obtained by Doron Zeilberger and then reformatted by Manuel Kauers (it uses Mathematica syntax):
Conjecture: ((-1 + n)^2*n*a[n])/4 + (n*(-16 + 38*n + 11*n^2)*a[1 + n])/16 +
(3/2 + (139*n)/16 + (29*n^2)/8 + (3*n^3)/16)*a[2 + n] +
(-21/4 - (51*n)/4 - (79*n^2)/16 - (5*n^3)/8)*a[3 + n] +
(-15/2 - n/8 + (5*n^2)/4 + n^3/8)*a[4 + n] +
(603/4 + (307*n)/4 + (49*n^2)/4 + (11*n^3)/16)*a[5 + n] +
(-41 - (533*n)/16 - (49*n^2)/8 - (5*n^3)/16)*a[6 + n] +
(-911/2 - 161*n - (303*n^2)/16 - (3*n^3)/4)*a[7 + n] +
(-363 - (417*n)/4 - (37*n^2)/4 - n^3/4)*a[8 + n] +
(-993/4 - 53*n - (11*n^2)/4)*a[9 + n] + (-130 - (93*n)/4 - n^2)*a[10 + n] +
(-71/4 - 2*n)*a[11 + n] + (-10 - n)*a[12 + n] + a[13 + n] == 0.
(End)
From Mark van Hoeij, Jul 25 2012: (Start)
A compact way to write the order 13 recurrence is as follows:
Let b(n) = a(n+3) + a(n+2) + (n/2+2)*a(n+1) + (n-1)*a(n)/2
and c(n) = b(n+4) + (n/2+2)*b(n+2) - b(n+1)/2 + (1-n)*b(n)/2;
then c(n+6) - (n+11)*c(n+5) - (2*n+75/4)*c(n+4) + (3-n)*c(n+3)/4 - c(n+2)/2 - (7*n+22)*c(n+1)/4-n*c(n) = 0. (End)

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

A189838 Number of ways to place n nonattacking composite pieces rook + rider[2,2] on an n X n chessboard.

Original entry on oeis.org

1, 2, 4, 16, 36, 128, 672, 4264, 25044, 173712, 1318904, 11069056, 96808692, 927478976, 9435033872, 103783608480, 1195155968388
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+2k)-p(j)|<>2k for all j>=1, k>=1, j+2k<=n

Links

Crossrefs

Cf. A110128, A000170

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

A117572 Expansion of (1 + 2*x^2)/((1 - x^2)*(1 - x^3)).

Original entry on oeis.org

1, 0, 3, 1, 3, 3, 4, 3, 6, 4, 6, 6, 7, 6, 9, 7, 9, 9, 10, 9, 12, 10, 12, 12, 13, 12, 15, 13, 15, 15, 16, 15, 18, 16, 18, 18, 19, 18, 21, 19, 21, 21, 22, 21, 24, 22, 24, 24, 25, 24, 27, 25, 27, 27, 28, 27, 30, 28, 30, 30, 31, 30, 33, 31, 33, 33, 34, 33, 36, 34, 36, 36, 37, 36, 39, 37
Offset: 0

Views

Author

Paul Barry, Mar 29 2006

Keywords

Comments

Diagonal sums of A110128 [this cross-reference is wrong - N. J. A. Sloane, Jan 01 2008]. Partial sums are A117573.

Links

Programs

  • Maxima
    a(n):=(n+1)/2+3/4*(-1)^n+1/12-(1/3)*(-2)^fix(mod(n,3)/2); /* Tani Akinari, Nov 10 2012 */

Formula

a(n) = a(n-2)+a(n-3)-a(n-5).
a(n) = cos(2*Pi*n/3+Pi/3)/3-sin(2*Pi*n/3+Pi/3)/sqrt(3)+3(-1)^n/4+(6n+7)/12.
a(n) = Sum_{k=0..floor(n/2)} 2*A001045(L((n-2k+2)/3)) where L(j/p) is the Legendre symbol of j and p.

A360384 Number of permutations p of [n] satisfying |p(i+7) - p(i)| <> 7 for all 1 <= i <= n-7.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 5040, 38880, 323520, 2953728, 29666304, 326584320, 3919011840, 50969640960, 705722941440, 10504707686400, 167258901086208, 2836455721721856, 51038126752727040, 971132188892405760
Offset: 0

Views

Author

Seiichi Manyama, Feb 05 2023

Keywords

Links

Crossrefs

Column k=7 of A333706.
Cf. A110128.

Extensions

a(12)-a(20) from Alois P. Heinz, Feb 07 2023
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