A061996 -id:A061996 - cp's OEIS frontend

A172202 Number of ways to place 3 nonattacking kings on a 3 X n board.

0, 0, 8, 34, 105, 248, 490, 858, 1379, 2080, 2988, 4130, 5533, 7224, 9230, 11578, 14295, 17408, 20944, 24930, 29393, 34360, 39858, 45914, 52555, 59808, 67700, 76258, 85509, 95480, 106198, 117690, 129983, 143104, 157080, 171938, 187705

Offset: 1

Vaclav Kotesovec, Jan 29 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Vaclav Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A047659, A061989, A061996.

[0] cat [(n-2)*(9*n^2-45*n+70)/2: n in [2..50]]; // G. C. Greubel, Apr 29 2022

CoefficientList[Series[x^2*(8+2*x+17*x^2)/(1-x)^4, {x, 0, 50}], x] (* Vincenzo Librandi, May 27 2013 *)
LinearRecurrence[{4,-6,4,-1},{0,0,8,34,105},40] (* Harvey P. Dale, Oct 07 2023 *)

[(1/8)*(n-2)*(9*(2*n-5)^2+55) +17*bool(n==1) for n in (1..50)] # G. C. Greubel, Apr 29 2022

a(n) = (n-2)*(9*n^2 - 45*n + 70)/2, n>=2.
G.f.: x^3*(8+2*x+17*x^2)/(1-x)^4. - Vaclav Kotesovec, Mar 24 2010
E.g.f.: 70 + 17*x + (1/2)*(-140 + 106*x - 36*x^2 + 9*x^3)*exp(x). - G. C. Greubel, Apr 29 2022

More terms from Vincenzo Librandi, May 27 2013

A194788 Number of ways to place 7 nonattacking kings on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 242, 51504, 2484382, 44601420, 450193818, 3112919712, 16471667554, 71393226972, 265069706646, 869583076752, 2577681275622, 7020477731884, 17794428237522, 42397762374912, 95726217156906, 206149749502012, 425731784898894, 846919172059632

Offset: 1

Andrew Woods, Sep 02 2011

nonn

V. Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes

Cf. A061995, A061996, A061997, A061998, A172158.

a(n) = (n^14 - 189n^12 + 252n^11 + 15211n^10 - 38640n^9 - 649215n^8 + 2408700n^7 + 14771764n^6 - 75856200n^5 - 144099396n^4 + 1198867488n^3 - 255900576n^2 - 7543005120n + 10617929280)/5040, n>=6. - Andrew Woods, Sep 02 2011

G.f.: 2*x^5*(1930*x^15 - 20052*x^14 + 87663*x^13 - 265681*x^12 + 816798*x^11 - 2117376*x^10 + 2865281*x^9 + 557737*x^8 - 6577818*x^7 + 3848604*x^6 + 8828017*x^5 - 9464319*x^4 - 6316750*x^3 - 868616*x^2 - 23937*x - 121)/ (x-1)^15. - Vaclav Kotesovec, Nov 06 2011

A172207 Number of ways to place 3 nonattacking bishops on a 3 X n board.

Original entry on oeis.org

1, 6, 26, 86, 211, 426, 758, 1234, 1881, 2726, 3796, 5118, 6719, 8626, 10866, 13466, 16453, 19854, 23696, 28006, 32811, 38138, 44014, 50466, 57521, 65206, 73548, 82574, 92311, 102786, 114026, 126058, 138909, 152606, 167176, 182646, 199043, 216394, 234726, 254066

Offset: 1

Vaclav Kotesovec, Jan 29 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A172124, A061989, A047659, A061996.

CoefficientList[Series[(2 x^6 + 14 x^3 + 8 x^2 + 2 x + 1) / (x - 1)^4, {x, 0, 50}], x] (* Vincenzo Librandi, May 27 2013 *)

a(n) = (9n^3 - 45n^2 + 106n - 108)/2, n>=4.

G.f.: x*(2*x^6+14*x^3+8*x^2+2*x+1)/(x-1)^4. - Vaclav Kotesovec, Mar 25 2010

A201369 Number of ways to place 8 nonattacking kings on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 27, 21792, 3324193, 119138166, 1979541332, 20142680752, 145977165234, 824771174978, 3850985758339, 15461577137802, 54912339921707, 176153338628674, 518569625849418, 1418340918023792, 3639736652346172, 8833161922947702, 20405252721413369

Offset: 1

Vaclav Kotesovec, Nov 30 2011

nonn

V. Kotesovec, Non-attacking chess pieces

Cf. A061995, A061996, A061997, A061998, A172158, A193580, A194788.

Explicit formula (Vaclav Kotesovec, after values computed by Andrew Woods, Nov 30 2011): (n^16 - 252*n^14 + 336*n^13 + 27762*n^12 - 70896*n^11 - 1699656*n^10 + 6330240*n^9 + 60677169*n^8 - 304864560*n^7 - 1181816748*n^6 + 8314366704*n^5 + 8495481308*n^4 - 121101870624*n^3 + 74007948336*n^2 + 730891869120*n - 1180990460160)/40320, n>=7.

G.f.: -x^5*(14882*x^18 - 180784*x^17 + 1061244*x^16 - 4500406*x^15 + 15038864*x^14 - 34328850*x^13 + 40903004*x^12 - 8667835*x^11 + 23857551*x^10 - 260744627*x^9 + 545801251*x^8 - 276255996*x^7 - 467674682*x^6 + 484515328*x^5 + 391528458*x^4 + 65572237*x^3 + 2957401*x^2 + 21333*x + 27)/(x-1)^17.

A194651 Number of ways to place 3 nonattacking kings on an n X n cylindrical chessboard.

Original entry on oeis.org

0, 0, 0, 88, 785, 3528, 11151, 28560, 63513, 127520, 236863, 413736, 687505, 1096088, 1687455, 2521248, 3670521, 5223600, 7286063, 9982840, 13460433, 17889256, 23466095, 30416688, 38998425, 49503168, 62260191, 77639240, 96053713, 117963960, 143880703

Offset: 1

Vaclav Kotesovec, Aug 31 2011

nonn

V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights
Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Cf. A061996, A179404, A194650.

CoefficientList[Series[-x^3*(15*x^6 - 89*x^5 + 196*x^4 - 140*x^3 - 119*x^2 + 169*x + 88)/(x - 1)^7, {x, 0, 30}], x] (* Wesley Ivan Hurt, Dec 27 2023 *)

a(n) = 1/6*n*(n^5 - 27*n^3 + 18*n^2 + 194*n - 228), n>=4.

G.f.: -x^4*(15*x^6 - 89*x^5 + 196*x^4 - 140*x^3 - 119*x^2 + 169*x + 88)/(x-1)^7.

A201771 Number of ways to place 9 nonattacking kings on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 1, 3600, 2882737, 229095676, 6655170642, 103395053720, 1051588999820, 7878155295948, 46838274976147, 232322652402464, 995789500001315, 3784235129731708, 12999197522073908, 40969826999523768, 119876498636101786, 328726265508168780, 851369417500529061

Offset: 1

Vaclav Kotesovec, Dec 04 2011

nonn

V. Kotesovec, Non-attacking chess pieces

Cf. A061995, A061996, A061997, A061998, A172158, A193580, A194788, A201369.

Explicit formula (Vaclav Kotesovec, after values computed by Andrew Woods, Dec 04 2011): n^18/362880 - n^16/1120 + n^15/840 + 1559*n^14/12096 - 119*n^13/360 - 7681*n^12/720 + 479*n^11/12 + 9383677*n^10/17280 - 195031*n^9/72 - 24176483*n^8/1440 + 4447749*n^7/40 + 5032857271*n^6/18144 - 495178813*n^5/180 - 2551293629*n^4/2520 + 1588223225*n^3/42 - 11469403819*n^2/315 - 664490248*n/3 + 405670140, n>=8.

G.f.: x^5*(54764*x^21 - 805588*x^20 + 6061268*x^19 - 31485512*x^18 + 117971558*x^17 - 312791986*x^16 + 620038858*x^15 - 1193322246*x^14 + 2685590901*x^13 - 4918483903*x^12 + 3824558880*x^11 + 5110355848*x^10 - 13987162841*x^9 + 5213745395*x^8 + 15789867458*x^7 - 14255103822*x^6 - 13342741937*x^5 - 2791816301*x^4 - 174938304*x^3 - 2814508*x^2 - 3581*x - 1)/(x-1)^19.

A220467 Number of ways to place 10 nonattacking kings on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1601292, 314949564, 17143061738, 423677826986, 6210264633994, 62831788827614, 481992723228798, 2982908737810114, 15548436178142582, 70420082692285198, 283631426534134042, 1034163399690010346, 3461457325296584554, 10754832937513676198

Offset: 1

Vaclav Kotesovec, Dec 15 2012

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Kotesovec, Non-attacking chess pieces

Cf. A061995 (2 kings), A061996 (3 kings), A061997 (4 kings).
Cf. A061998 (5 kings), A172158 (6 kings), A194788 (7 kings).
Cf. A201369 (8 kings), A201771 (9 kings).
Column k=10 of A193580.

Rest[CoefficientList[Series[-2*x^7*(97581*x^22 - 1758956*x^21 + 16320562*x^20 - 100734462*x^19 + 443795293*x^18 - 1471049082*x^17 + 3971393292*x^16 - 9304893422*x^15 + 17917931016*x^14 - 22612415810*x^13 + 6949925614*x^12 + 21430418050*x^11 + 9738010368*x^10 - 153051533038*x^9 + 256884162558*x^8 - 71451647970*x^7 - 265785285277*x^6 + 220345759446*x^5 + 251887022384*x^4 + 63841610284*x^3 + 5432696107*x^2 + 140661216*x + 800646)/(x-1)^21, {x, 0, 20}], x]]

a(n) = n^20/3628800 - n^18/8960 + n^17/6720 + 353*n^16/17280 - 53*n^15/1008 - 29467*n^14/13440 + 11867*n^13/1440 + 25901053*n^12/172800 - 107495*n^11/144 - 8467959*n^10/1280 + 122792641*n^9/2880 + 32499630031*n^8/181440 - 112903333*n^7/72 - 16042907329*n^6/6720 + 36445613711*n^5/1008 - 1784819159*n^4/300 - 9997453897*n^3/21 + 85979117831*n^2/140 + 13635070421*n/5 - 5609601346, for n>=9.

G.f.: -2*x^7*(97581*x^22 - 1758956*x^21 + 16320562*x^20 - 100734462*x^19 + 443795293*x^18 - 1471049082*x^17 + 3971393292*x^16 - 9304893422*x^15 + 17917931016*x^14 - 22612415810*x^13 + 6949925614*x^12 + 21430418050*x^11 + 9738010368*x^10 - 153051533038*x^9 + 256884162558*x^8 - 71451647970*x^7 - 265785285277*x^6 + 220345759446*x^5 + 251887022384*x^4 + 63841610284*x^3 + 5432696107*x^2 + 140661216*x + 800646)/(x-1)^21.

A226997 Irregular triangle read by rows: T(n,k) is the number of distinct tilings by squares of an n X n square lattice that contain k nodes unconnected to any of their neighbors.

Original entry on oeis.org

1, 1, 1, 1, 4, 0, 0, 1, 1, 9, 16, 8, 5, 0, 0, 0, 0, 1, 1, 16, 78, 140, 88, 44, 68, 32, 0, 4, 0, 0, 0, 0, 0, 0, 1, 1, 25, 228, 964, 2003, 2178, 1842, 1626, 725, 290, 376, 184, 140, 76, 4, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 36, 520, 3920, 16859, 42944, 67312

Offset: 1

Christopher Hunt Gribble, Jun 26 2013

The n-th row contains (n-1)^2 + 1 elements.

			For n = 3, there are 4 tilings that contain 1 isolated node, so T(3,1) = 4. A 2 X 2 square contains 1 isolated node.  Consider that each tiling is composed of ones and zeros where a one represents a node with one or more links to its neighbors and a zero represents a node with no links to its neighbors.  Then the 4 tilings are:
1 1 1 1    1 1 1 1    1 1 1 1    1 1 1 1
1 0 1 1    1 1 0 1    1 1 1 1    1 1 1 1
1 1 1 1    1 1 1 1    1 0 1 1    1 1 0 1
1 1 1 1    1 1 1 1    1 1 1 1    1 1 1 1
The irregular triangle begins:
\ k 0     1     2     3     4     5     6     7     8     9  ...
n
1   1
2   1     1
3   1     4     0     0     1
4   1     9    16     8     5     0     0     0     0     1
5   1    16    78   140    88    44    68    32     0     4  ...
6   1    25   228   964  2003  2178  1842  1626   725   290  ...
7   1    36   520  3920 16859 42944 67312 72980 69741 62952  ...

Alois P. Heinz, Rows n = 1..16, flattened (Rows n = 1..7 from Christopher Hunt Gribble)

Cf. A045846.

b:= proc(n, l) option remember; local i, k, s, t;
      if max(l[])>n then 0 elif n=0 or l=[] then 1
    elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l))
    else for k do if l[k]=0 then break fi od; s:=0;
         for i from k to nops(l) while l[i]=0 do s:=s+x^((i-k)^2)
          *b(n, [l[j]$j=1..k-1, 1+i-k$j=k..i, l[j]$j=i+1..nops(l)])
         od; expand(s)
      fi
    end:
T:= n-> (l-> seq(coeff(l,x,i), i=0..degree(l)))(b(n, [0$n])):
seq(T(n), n=1..9);  # Alois P. Heinz, Jun 27 2013

Sum_{k=0..(n-1)^2} T(n,k) = A045846(n).
From Christopher Hunt Gribble, Jul 02 2013: (Start)
It appears that:
T(n,1) = (n-1)^2, n>1 = A000290(n-1).
T(n,2) = (n-2)(n-3)(n^2+n-4)/2, n>2 = A061995(n-1).
T(n,3) = (n-2)(n-3)(n^4-n^3-23n^2+15n+140)/6, n>2 = A061996(n-1).
T(n,4) = (n^8 - 8n^7 - 26*n^6 + 340*n^5 - 105*n^4 - 4708*n^3 + 6814*n^2 + 20852*n - 40248)/24, n>3. (End)

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A172202 Number of ways to place 3 nonattacking kings on a 3 X n board.

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A194788 Number of ways to place 7 nonattacking kings on an n X n board.

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A172207 Number of ways to place 3 nonattacking bishops on a 3 X n board.

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A201369 Number of ways to place 8 nonattacking kings on an n X n board.

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A194651 Number of ways to place 3 nonattacking kings on an n X n cylindrical chessboard.

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A201771 Number of ways to place 9 nonattacking kings on an n X n board.

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A220467 Number of ways to place 10 nonattacking kings on an n X n board.

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A226997 Irregular triangle read by rows: T(n,k) is the number of distinct tilings by squares of an n X n square lattice that contain k nodes unconnected to any of their neighbors.

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