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 1-4 of 4 results.

A201511 Number of ways to place n nonattacking wazirs on an n X n board.

Original entry on oeis.org

1, 1, 2, 22, 405, 10741, 368868, 15516804, 771464278, 44218721793, 2868879752822, 207739939478618, 16602826428818482, 1451305771147909684, 137715836041691050398, 14096224186664736126206, 1547966111897855935957132, 181519663430661533452513680, 22636566614411901986006002896
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 02 2011

Keywords

Comments

Wazir is a leaper [0,1].

Links

Crossrefs

Cf. A172225, A172226, A172227, A172228, A178409, A201507, A201508, A201510.
Cf. A006506.
Main diagonal of A232833.

Formula

Asymptotics (Vaclav Kotesovec, Nov 29 2011): a(n) ~ n^(2n)/n!*exp(-5/2).

Extensions

a(19)-a(20) from Vaclav Kotesovec, Aug 30 2016
a(0)=1 prepended by Alois P. Heinz, Apr 16 2024

A232833 Triangle read by rows: T(n,k) = number of n X n binary matrices with k pairwise nonadjacent 1's, n >= 0, k = 0..n^2.

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 0, 0, 1, 9, 24, 22, 6, 1, 0, 0, 0, 0, 1, 16, 96, 276, 405, 304, 114, 20, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 25, 260, 1474, 5024, 10741, 14650, 12798, 7157, 2578, 618, 106, 14, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 36, 570, 5248, 31320, 127960, 368868
Offset: 0

Views

Author

Heinrich Ludwig, Dec 01 2013

Keywords

Comments

Also number of ways to place k non-attacking wazirs on an n X n board.
Two matrix elements are considered adjacent if the difference of their row indices is 1 and the column indices are equal, or vice versa (von Neumann neighborhood).
If only non-equivalent (mod D_4) matrices are counted, the corresponding numbers are given by A232569.
Rows with trailing zeros dropped give the coefficients of the independence polynomial for the n X n grid graph. - Eric W. Weisstein, May 31 2017

Examples

			Triangle begins:
  1;
  1,  1;
  1,  4,  2,   0,   0;
  1,  9, 24,  22,   6,   1,   0,  0, 0, 0;
  1, 16, 96, 276, 405, 304, 114, 20, 2, 0, 0, 0, 0, 0, 0, 0, 0;
  ...

Links

Crossrefs

Cf. A232569, A006506 (row sums).
Main diagonal gives A201511.

Programs

  • Maple
    b:= proc(n, l) option remember; local k;
      if n=0 then 1
    elif min(l)>0 then b(n-1, map(x-> x-1, l))
    else for k while l[k]>0 do od;
         b(n, subsop(k=1, l))+expand(x*`if`(n>0, `if`(k (p-> seq(coeff(p,x,i), i=0..n^2))(b(n, [0$n])):
seq(T(n), n=0..6);  # Alois P. Heinz, Apr 16 2024
  • Mathematica
    b[n_, l_] := b[n, l] = Module[{k},
   Which[n == 0, 1,
   Min[l] > 0, b[n - 1, l - 1],
   True, For[k = 1, l[[k]] > 0, k++];
      b[n, ReplacePart[l, k -> 1]] + Expand[x*If[n > 0, If[k < Length[l],
      b[n, ReplacePart[l, {k -> 2, k + 1 -> 1}]],
      b[n, ReplacePart[l, k -> 2]], 0]]]]];
T[n_] := With[{p = b[n, Table[0, {n}]]}, Table[Coefficient[p, x, i], {i, 0, n^2}]]
Table[T[n], {n, 0, 6}] // Flatten (* Jean-François Alcover, Aug 09 2024, after Alois P. Heinz *)

Formula

T(n,0) = A000012(n);
T(n,1) = A000290(n), n >= 1;
T(n,2) = A172225(n), n >= 2;
T(n,3) = A172226(n), n >= 2;
T(n,4) = A172227(n), n >= 2;
T(n,5) = A172228(n), n >= 3;
T(n,6) = A178409(n), n >= 3;
T(n,7) = A201507(n), n >= 3;
T(n,8) = A201508(n), n >= 3;
T(n,9) = A201510(n), n >= 3;

Extensions

T(0,0)=1 inserted by Alois P. Heinz, Apr 16 2024

A201508 Number of ways to place 8 nonattacking wazirs on an n X n board.

Original entry on oeis.org

0, 0, 0, 2, 7157, 1143638, 44031035, 771464278, 8219304992, 62114308624, 364798895986, 1765597908290, 7329246973785, 26849172347850, 88645482921449, 268042562131202, 751635857876290, 1974215715426992, 4896315981217168, 11542835604897814, 26008912447737323
Offset: 1

Views

Author

Vaclav Kotesovec, Dec 02 2011

Keywords

Comments

Wazir is a leaper [0,1].

Links

Crossrefs

Cf. A172225, A172226, A172227, A172228, A178409, A201507.

Formula

Explicit formula: n^16/40320 - n^14/288 + n^13/360 + 623*n^12/2880 - 41*n^11/120 - 5521*n^10/720 + 649*n^9/36 + 941767*n^8/5760 - 12485*n^7/24 - 577177*n^6/288 + 3102289*n^5/360 + 12378183*n^4/1120 - 1545219*n^3/20 + 1588751*n^2/120 + 581605*n/2 - 308806, n>=7.
G.f.: -x^4*(12*x^19 - 122*x^18 + 1130*x^17 - 6776*x^16 + 11180*x^15 + 33894*x^14 + 82772*x^13 - 1938093*x^12 + 7575029*x^11 - 10487057*x^10 - 11993287*x^9 + 70715064*x^8 - 109013258*x^7 + 41757444*x^6 + 331980470*x^5 + 173609451*x^4 + 25561181*x^3 + 1022241*x^2 + 7123*x + 2)/(x-1)^17.
a(n) = A232833(n,8). - R. J. Mathar, Apr 11 2024

A201510 Number of ways to place 9 nonattacking wazirs on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 2578, 1247116, 97284860, 2817340064, 44218721793, 457851259868, 3506596268191, 21355746900992, 108582220087480, 477032549147428, 1857084405493128, 6529640029479296, 21044674478336823, 62903854631232636, 176034055470126073, 464793685059669728
Offset: 1

Views

Author

Vaclav Kotesovec, Dec 02 2011

Keywords

Comments

Wazir is a leaper [0,1].

Links

Crossrefs

Cf. A172225, A172226, A172227, A172228, A178409, A201507, A201508.

Formula

Explicit formula: n^18/362880 - n^16/2016 + n^15/2520 + 349*n^14/8640 - 23*n^13/360 - 277*n^12/144 + 163*n^11/36 + 199529*n^10/3456 - 4381*n^9/24 - 313811*n^8/288 + 1622087*n^7/360 + 1073654363*n^6/90720 - 12207881*n^5/180 - 24979477*n^4/504 + 72278641*n^3/126 - 11491519*n^2/45 - 6271604*n/3 + 2530368, n>=8.
G.f.: x^5*(14*x^21 - 226*x^20 + 2514*x^19 - 15414*x^18 + 54363*x^17 - 241813*x^16 + 1440666*x^15 - 4412622*x^14 - 2699713*x^13 + 64333547*x^12 - 202456488*x^11 + 209746960*x^10 + 407620979*x^9 - 1743413585*x^8 + 2469587594*x^7 - 1465834094*x^6 - 9995512037*x^5 - 6126508561*x^4 - 1179686478*x^3 - 74030494*x^2 - 1198134*x - 2578)/(x-1)^19.
a(n) = A232833(n,9). - R. J. Mathar, Apr 11 2024
Showing 1-4 of 4 results.

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