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.

A278688 Triangle read by rows T(n, k) = number of non-equivalent ways to place k non-attacking ferses on an n X n board.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 6, 7, 6, 2, 1, 1, 3, 17, 45, 92, 99, 76, 27, 7, 1, 6, 43, 225, 832, 2102, 3773, 4860, 4643, 3356, 1868, 795, 248, 56, 8, 1, 1, 6, 84, 709, 4500, 19987, 66201, 164423, 314224, 465230, 540247, 492206, 352300, 195717, 83247, 26083, 5754, 780, 55
Offset: 1

Views

Author

Heinrich Ludwig, Nov 27 2016

Keywords

Comments

The triangle T(n, k) is irregularly shaped: 0 <= k <= A093005(n), which means that A093005(n) is the maximal number of non-attacking ferses that can be placed on an n X n board. First row corresponds to n = 1. First column corresponds to k = 0.
Two placements that differ by rotation or reflection are counted only once.
A fers is a fairy chess piece attacking one step ne-nw-sw-se.

Examples

			Triangle begins:
1, 1;
1, 1,  1;
1, 3,  6,  7,  6,  2,  1;
1, 3, 17, 45, 92, 99, 76, 27, 7;

Links

Crossrefs

Cf. A008805, A232567, A278682, A278683, A278684, A278685, A278686, (columns 2 through 8 of this sequence, respectively), A278687, A093005 (row length - 1).

A278682 Number of non-equivalent ways to place 3 non-attacking ferses on an n X n board.

Original entry on oeis.org

0, 0, 7, 45, 225, 709, 1974, 4524, 9614, 18382, 33425, 56895, 93447, 146715, 224280, 331814, 480844, 679724, 945099, 1288737, 1733725, 2296065, 3006762, 3886960, 4977210, 6304794, 7921589, 9862099, 12191459, 14952567, 18225900, 22064010, 26564952, 31792280
Offset: 1

Views

Author

Heinrich Ludwig, Nov 26 2016

Keywords

Comments

A fers is a leaper [1, 1].
Rotations and reflections of placements are not counted. If they are to be counted, see A201244.

Examples

			There are 7 ways to place 3 non-attacking ferses "X" on a 3 X 3 board, rotations and reflections being ignored
   XXX   XX.   X.X   ...   X..   X..   X..
   ...   ...   ...   XXX   X.X   ...   ...
   ...   ..X   .X.   ...   ...   XX.   X.X

Links

Crossrefs

Cf. A201244, A232567 (2 ferses), A278683 (4 ferses), A278684 (5 ferses), A278685 (6 ferses), A278686 (7 ferses), A278687, A278688.

Programs

  • Mathematica
    Table[Boole[n > 2] ((n^6 - 15 n^4 + 32 n^3 + 14 n^2 - 116 n + 96) + Boole[OddQ@ n] (8 n^3 - 9 n^2 - 20 n + 9))/48, {n, 34}] (* Michael De Vlieger, Nov 30 2016 *)
  • PARI
    concat(vector(2), Vec(x^3*(7 + 24*x + 83*x^2 + 66*x^3 + 75*x^4 - 15*x^6 - 2*x^7 + 2*x^8) / ((1 - x)^7*(1 + x)^4) + O(x^40))) \\ Colin Barker, Dec 07 2016

Formula

a(n) = ((n^6 - 15*n^4 + 32*n^3 + 14*n^2 - 116*n + 96) + IF(MOD(n, 2) = 1, 8*n^3 - 9*n^2 - 20*n + 9))/48.
a(n) = 3*a(n-1) + a(n-2) - 11*a(n-3) + 6*a(n-4) + 14*a(n-5) - 14*a(n-6) - 6*a(n-7) + 11*a(n-8) - a(n-9) - 3*a(n-10) + a(n-11).
From Colin Barker, Dec 07 2016: (Start)
a(n) = (n^6 - 15*n^4 + 32*n^3 + 14*n^2 - 116*n + 96)/48 for n even.
a(n) = (n^6 - 15*n^4 + 40*n^3 + 5*n^2 - 136*n + 105)/48 for n odd.
G.f.: x^3*(7 + 24*x + 83*x^2 + 66*x^3 + 75*x^4 - 15*x^6 - 2*x^7 + 2*x^8) / ((1 - x)^7*(1 + x)^4).
(End)

A278683 Number of non-equivalent ways to place 4 non-attacking ferses on an n X n board.

Original entry on oeis.org

0, 0, 6, 92, 832, 4500, 18229, 58881, 163509, 401259, 898420, 1861146, 3625546, 6694982, 11829267, 20099815, 33036079, 52700901, 81916834, 124362664, 184907220, 269726216, 386776561, 545930397, 759628777, 1043027055, 1414873104, 1897655046, 2518755934, 3310591194
Offset: 1

Views

Author

Heinrich Ludwig, Nov 26 2016

Keywords

Comments

A fers is a leaper [1, 1].
Rotations and reflections of placements are not counted. If they are to be counted, see A201245.

Examples

			There are 6 ways to place 4 non-attacking ferses on a 3 X 3 board rotations and reflections being ignored:
   XXX   XXX   X.X   X.X   XX.   XX.
   ...   ...   ...   ...   ...   ...
   ..X   .X.   X.X   XX.   XX.   .XX

Links

Crossrefs

Cf. A201245, A232567 (2 ferses), A278682 (3 ferses), A278684 (5 ferses), A278685 (6 ferses), A278686 (7 ferses), A278687, A278688.

Programs

  • Mathematica
    Table[Boole[n > 2] (n^8 - 30 n^6 + 48 n^5 + 328 n^4 - 1056 n^3 - 200 n^2 + 4176 n - 4032 + Boole[OddQ@ n] (14 n^4 - 48 n^3 - 38 n^2 + 336 n - 459))/192, {n, 30}] (* Michael De Vlieger, Nov 30 2016 *)
  • PARI
    concat(vector(2), Vec(x^3*(6 +68*x +470*x^2 +1360*x^3 +2419*x^4 +1909*x^5 +836*x^6 -232*x^7 -192*x^8 +30*x^9 +54*x^10 -9*x^12 +x^13) / ((1 -x)^9*(1 +x)^5) + O(x^40))) \\ Colin Barker, Dec 10 2016

Formula

a(n) = (n^8 - 30*n^6 + 48*n^5 + 328*n^4 - 1056*n^3 - 200*n^2 + 4176*n - 4032 + IF(MOD(n, 2) = 1, 14*n^4 - 48*n^3 - 38*n^2 + 336*n - 459))/192 for n>=3.
a(n) = 4*a(n-1) - a(n-2) - 16*a(n-3) + 19*a(n-4) + 20*a(n-5) - 45*a(n-6) + 45*a(n-8) - 20*a(n-9) - 19*a(n-10) + 16*a(n-11) + a(n-12) - 4*a(n-13) + a(n-14) for n>=17.
G.f.: x^3*(6 +68*x +470*x^2 +1360*x^3 +2419*x^4 +1909*x^5 +836*x^6 -232*x^7 -192*x^8 +30*x^9 +54*x^10 -9*x^12 +x^13) / ((1 -x)^9*(1 +x)^5). - Colin Barker, Dec 10 2016

A278684 Number of non-equivalent ways to place 5 non-attacking ferses on an n X n board.

Original entry on oeis.org

0, 0, 2, 99, 2102, 19987, 124676, 571418, 2122841, 6704061, 18711691, 47235845, 109938296, 238950999, 490309398, 957267228, 1790325363, 3224010105, 5615368229, 9493358359, 15627413290, 25112609019, 39484650296, 60859027054, 92114682749, 137111560949, 200972392655
Offset: 1

Views

Author

Heinrich Ludwig, Nov 26 2016

Keywords

Comments

A fers is a leaper [1, 1].
Rotations and reflections of placements are not counted. If they are to be counted, see A201246.

Examples

			There are 2 ways to place 5 non-attacking ferses on a 3 X 3 board, rotations and reflections being ignored:
   XXX   XXX
   ...   ...
   X.X   XX.

Links

Crossrefs

Cf. A201246, A232567 (2 ferses), A278682 (3 ferses), A278683 (4 ferses), A278685 (6 ferses), A278686 (7 ferses), A278687, A278688.

Programs

  • Magma
    [0, 0, 2] cat [(n^10 - 50*n^8 + 80*n^7 + 955*n^6 - 2828*n^5 - 7090*n^4 + 36860*n^3 - 10856*n^2 - 133712*n + 161280 + ((1-(-1)^n)/2)*(52*n^5 - 145*n^4 - 580*n^3 + 2320*n^2 - 1152*n - 15))/960 : n in [4..30]]; // Wesley Ivan Hurt, Nov 27 2016
  • Maple
    A278684:=n->(n^10 - 50*n^8 + 80*n^7 + 955*n^6 - 2828*n^5 - 7090*n^4 + 36860*n^3 - 10856*n^2 - 133712*n + 161280 + ((1-(-1)^n)/2)*(52*n^5 - 145*n^4 - 580*n^3 + 2320*n^2 - 1152*n - 15))/960: 0, 0, 2, seq(A278684(n), n=4..30); # Wesley Ivan Hurt, Nov 27 2016
  • Mathematica
    Join[{0, 0, 2}, Table[(n^10 - 50*n^8 + 80*n^7 + 955*n^6 - 2828*n^5 - 7090*n^4 + 36860*n^3 - 10856*n^2 - 133712*n + 161280 + ((1-(-1)^n)/2)*(52*n^5 - 145*n^4 - 580*n^3 + 2320*n^2 - 1152*n - 15))/960, {n, 4, 30}]] (* Wesley Ivan Hurt, Nov 27 2016 *)
  • PARI
    concat(vector(2), Vec(x^3*(2 +89*x +1615*x^2 +9913*x^3 +35049*x^4 +66034*x^5 +78731*x^6 +45748*x^7 +9902*x^8 -5540*x^9 -1343*x^10 +1685*x^11 +409*x^12 -334*x^13 -83*x^14 +38*x^15 +6*x^16 -x^17) / ((1 -x)^11*(1 +x)^6) + O(x^40))) \\ Colin Barker, Dec 10 2016

Formula

a(n) = (n^10 - 50*n^8 + 80*n^7 + 955*n^6 - 2828*n^5 - 7090*n^4 + 36860*n^3 - 10856*n^2 - 133712*n + 161280 + ((1-(-1)^n)/2)*(52*n^5 - 145*n^4 - 580*n^3 + 2320*n^2 - 1152*n - 15))/960 for n >= 4.
a(n) = 5*a(n-1) - 4*a(n-2) - 20*a(n-3) + 40*a(n-4) + 16*a(n-5) - 100*a(n-6) + 44*a(n-7) + 110*a(n-8) - 110*a(n-9) - 44*a(n-10) + 100*a(n-11) - 16*a(n-12) - 40*a(n-13) + 20*a(n-14) + 4*a(n-15) - 5*a(n-16) + a(n-17) for n >= 21.
G.f.: x^3*(2 +89*x +1615*x^2 +9913*x^3 +35049*x^4 +66034*x^5 +78731*x^6 +45748*x^7 +9902*x^8 -5540*x^9 -1343*x^10 +1685*x^11 +409*x^12 -334*x^13 -83*x^14 +38*x^15 +6*x^16 -x^17) / ((1 -x)^11*(1 +x)^6). - Colin Barker, Dec 10 2016

A278686 Number of non-equivalent ways to place 7 non-attacking ferses on an n X n board.

Original entry on oeis.org

0, 0, 0, 27, 4860, 164423, 2651890, 25981150, 180378380, 971905679, 4316504623, 16457726539, 55463445891, 168782705327, 471407278652, 1223710587908, 2982045310010
Offset: 1

Views

Author

Heinrich Ludwig, Dec 02 2016

Keywords

Comments

A fers is a leaper [1, 1].
Rotations and reflections of placements are not counted. If they are to be counted, see A201248.

Examples

			There are 27 non-equivalent ways to place 7 non-attacking ferses (X) on a 4 X 4 board, rotations and reflections being ignored, e.g., these two:
   XXXX   X.XX
   ....   ....
   XXX.   X.X.
   ....   X.X.

Links

Crossrefs

Cf. A201248, A232567 (2 ferses), A278682 (3 ferses), A278683 (4 ferses), A278684 (5 ferses), A278685 (6 ferses), A278687, A278688.
Showing 1-5 of 5 results.

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