A047461 -id:A047461 - cp's OEIS frontend

A047470 Numbers that are congruent to {0, 3} mod 8.

0, 3, 8, 11, 16, 19, 24, 27, 32, 35, 40, 43, 48, 51, 56, 59, 64, 67, 72, 75, 80, 83, 88, 91, 96, 99, 104, 107, 112, 115, 120, 123, 128, 131, 136, 139, 144, 147, 152, 155, 160, 163, 168, 171, 176, 179, 184, 187, 192, 195, 200, 203, 208, 211, 216, 219, 224, 227, 232

Offset: 1

N. J. A. Sloane

Maximum number of squares attacked by a queen on an n X n chessboard. - Stewart Gordon, Mar 23 2001
Equivalently, maximum vertex degree in the n X n queen graph. - Eric W. Weisstein, Jun 20 2017
Number of squares attacked by a queen on a toroidal chessboard. - Diego Torres (torresvillarroel(AT)hotmail.com), May 19 2001
List of squared distances between points of diamond 'lattice' with minimal distance sqrt(3). - Arnold Neumaier (Arnold.Neumaier(AT)univie.ac.at), Aug 01 2003
Draw a figure-eight knot diagram on the plane and assign a list of nonnegative numbers at each crossing as follows. Start with 0 and choose a crossing on the knot. Pick a direction and walk around the knot, appending the following nonnegative number everytime a crossing is visited. Two series of sequences are obtained: this sequence, A047535, A047452, A047617 and A047615, A047461, A047452, A047398 (see example). - Franck Maminirina Ramaharo, Jul 22 2018

			From _Franck Maminirina Ramaharo_, Jul 22 2018: (Start)
Consider the following equivalent figure-eight knot diagrams:
+---------------------+           +-----------------n
|                     |           |                 |
|           +---------B-----+     |           w-----A---e
|           |         |     |     |           |     |   |
|     n-----C---+     |     |     |           |     |   |
|     |     |   |     |     | <=> |   +-------B-----s   |
|     |     +---D-----+     |     |   |       |         |
|     |         |           |     |   |       |         |
w-----A---------e           |     +---C-------D---------+
      |                     |         |       |
      s---------------------+         +-------+
Uppercases A,B,C,D denote crossings, and lowercases n,s,w,e denote directions. Due to symmetry and ambient isotopy, all possible sequences are obtained by starting from crossing A and choose either direction 'n' or 's'.
Direction 'n':
A: 0, 3,  8, 11, 16, 19, 24, 27, 32, 35, 40, ... (this sequence);
B: 4, 7, 12, 15, 20, 23, 28, 31, 36, 39, 44, ... A047535;
C: 1, 6,  9, 14, 17, 22, 25, 30, 33, 38, 41, ... A047452;
D: 2, 5, 10, 13, 18, 21, 26, 29, 34, 37, 42, ... A047617.
Direction 's':
A: 0, 5,  8, 13, 16, 21, 24, 29, 32, 37, 40, ... A047615;
B: 1, 4,  9, 12, 17, 20, 25, 28, 33, 36, 41, ... A047461;
C: 2, 7, 10, 15, 18, 23, 26, 31, 34, 39, 42, ... A047524;
D: 3, 6, 11, 14, 19, 22, 27, 30, 35, 38, 43, ... A047398.
(End)

Muniru A Asiru, Table of n, a(n) for n = 1..5000
Eric Weisstein's World of Mathematics, Maximum Vertex Degree.
Eric Weisstein's World of Mathematics, Queen Graph.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A042948, A047398, A047461, A047452, A047524, A047535, A047615, A047617.

a:=[0,3,8];; for n in [4..50] do a[n]:=a[n-1]+a[n-2]-a[n-3]; od; a; # Muniru A Asiru, Jul 23 2018

a:=n->add(4+(-1)^j,j=1..n):seq(a(n),n=0..64); # Zerinvary Lajos, Dec 13 2008

With[{c = 8 Range[0, 30]}, Sort[Join[c, c + 3]]] (* Harvey P. Dale, Oct 11 2011 *)
Table[(8 n - 9 - (-1)^n)/2, {n, 20}] (* Eric W. Weisstein, Jun 20 2017 *)
LinearRecurrence[{1, 1, -1}, {0, 3, 8}, 20] (* Eric W. Weisstein, Jun 20 2017 *)
CoefficientList[Series[(x (3 + 5 x))/((-1 + x)^2 (1 + x)), {x, 0, 20}], x]  (* Eric W. Weisstein, Jun 20 2017 *)

forstep(n=0,200,[3,5],print1(n", ")) \\ Charles R Greathouse IV, Oct 17 2011

def A047470(n): return (n-1<<2)-(n&1^1) # Chai Wah Wu, Mar 30 2024

a(n) = a(n-1) + 4 + (-1)^n.
a(n) = a(n-1) + a(n-2) - a(n-3).
a(n) = A042948(n) + A005843(n).
G.f.: (3x+5*x^2)/((1-x)*(1-x^2)).
a(n) = 8*n - a(n-1) - 13 (with a(1)=0). - Vincenzo Librandi, Aug 06 2010
a(n+1) = Sum_{k>=0} A030308(n,k)*A171497(k). - Philippe Deléham, Oct 17 2011
a(n) = 4*n -(9 + (-1)^n)/2. - Arkadiusz Wesolowski, Sep 18 2012
E.g.f: (10 - exp(-x) + (8*x - 9)*exp(x))/2. - Franck Maminirina Ramaharo, Jul 22 2018
Sum_{n>=2} (-1)^n/a(n) = (sqrt(2)-1)*Pi/16 + log(2)/2 - sqrt(2)*log(sqrt(2)+1)/8. - Amiram Eldar, Dec 11 2021

More terms from Vincenzo Librandi, Aug 06 2010

A047452 Numbers that are congruent to {1, 6} mod 8.

Original entry on oeis.org

1, 6, 9, 14, 17, 22, 25, 30, 33, 38, 41, 46, 49, 54, 57, 62, 65, 70, 73, 78, 81, 86, 89, 94, 97, 102, 105, 110, 113, 118, 121, 126, 129, 134, 137, 142, 145, 150, 153, 158, 161, 166, 169, 174, 177, 182, 185, 190

Offset: 1

N. J. A. Sloane

Except for 1, numbers whose binary reflected Gray code (A014550) ends with 01. - Amiram Eldar, May 17 2021

Muniru A Asiru, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A014550, A047398, A047461, A047470, A047524, A047535, A047615, A047617.

Filtered([0..250], n->n mod 8=1 or n mod 8=6); # Muniru A Asiru, Jul 24 2018

seq(coeff(series(factorial(n)*((4+exp(-x)+(8*x-5)*exp(x))/2), x,n+1),x,n),n=1..60); # Muniru A Asiru, Jul 24 2018

Table[(8 n - 5 + (-1)^n)/2, {n, 1, 100}] (* Franck Maminirina Ramaharo, Jul 23 2018 *)
CoefficientList[ Series[(2x^2 + 5x + 1)/((x - 1)^2 (x + 1)), {x, 0, 50}], x] (* or *)
LinearRecurrence[{1, 1, -1}, {1, 6, 9}, 51] (* Robert G. Wilson v, Jul 24 2018 *)

makelist((8*n - 5 + (-1)^n)/2, n, 1, 100); /* Franck Maminirina Ramaharo, Jul 23 2018 */

def A047452(n): return (n<<2)-2-(n&1) # Chai Wah Wu, Mar 30 2024

G.f.: x*(1+5*x+2*x^2) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Dec 07 2011
E.g.f.: (4 + exp(-x) + (8*x - 5)*exp(x))/2. - Ilya Gutkovskiy, May 25 2016
a(n) = A047615(n) + 1. - Franck Maminirina Ramaharo, Jul 23 2018
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)+2)*Pi/16 + log(2)/8 + sqrt(2)*log(sqrt(2)+1)/8. - Amiram Eldar, Dec 11 2021

A047617 Numbers that are congruent to {2, 5} mod 8.

Original entry on oeis.org

2, 5, 10, 13, 18, 21, 26, 29, 34, 37, 42, 45, 50, 53, 58, 61, 66, 69, 74, 77, 82, 85, 90, 93, 98, 101, 106, 109, 114, 117, 122, 125, 130, 133, 138, 141, 146, 149, 154, 157, 162, 165, 170, 173, 178, 181, 186, 189, 194, 197, 202, 205, 210, 213, 218, 221, 226, 229, 234

Offset: 1

N. J. A. Sloane

Numbers whose binary reflected Gray code (A014550) ends with 11. - Amiram Eldar, May 17 2021

Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A014550, A047398, A047452, A047461, A047470, A047524, A047535, A047615.

Select[Range[300],MemberQ[{2,5},Mod[#,8]]&] (* or *) LinearRecurrence[ {1,1,-1},{2,5,10},80] (* Harvey P. Dale, Feb 23 2016 *)

makelist(4*n -(5 + (-1)^n)/2, n, 1, 100); /* Franck Maminirina Ramaharo, Jul 22 2018 */

def A047617(n): return (n-1<<2)+1+(n&1) # Chai Wah Wu, Mar 30 2024

a(n) = 8*n - a(n-1) - 9 (with a(1)=2). - Vincenzo Librandi, Aug 06 2010
a(n) = 4*n - (5 + (-1)^n)/2. - Arkadiusz Wesolowski, Sep 18 2012
G.f.: (2+3*x+3*x^2)/((-1+x)^2*(1+x)). - Harvey P. Dale, Feb 23 2016
a(1)=2, a(2)=5, a(3)=10, a(n) = a(n-1) + a(n-2) - a(n-3). - Harvey P. Dale, Feb 23 2016
From Franck Maminirina Ramaharo, Jul 22 2018: (Start)
a(n) = A047470(n) + 2.
E.g.f.: (6 - exp(-x) + (8*x - 5)*exp(x))/2. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(2)*Pi/16 - log(2)/8 + sqrt(2)*log(sqrt(2)+1)/8. - Amiram Eldar, Dec 11 2021

More terms from Vincenzo Librandi, Aug 06 2010

A047524 Numbers that are congruent to {2, 7} mod 8.

Original entry on oeis.org

2, 7, 10, 15, 18, 23, 26, 31, 34, 39, 42, 47, 50, 55, 58, 63, 66, 71, 74, 79, 82, 87, 90, 95, 98, 103, 106, 111, 114, 119, 122, 127, 130, 135, 138, 143, 146, 151, 154, 159, 162, 167, 170, 175, 178, 183, 186, 191, 194, 199, 202, 207, 210, 215, 218, 223, 226, 231, 234

Offset: 1

N. J. A. Sloane

A195605 is a subsequence. - Bruno Berselli, Sep 21 2011

Muniru A Asiru, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A047398, A047461, A047452, A047470, A047535, A047615, A047617, A195605.

Filtered([0..250],n->n mod 8=2 or n mod 8=7); # Muniru A Asiru, Aug 06 2018

seq(coeff(series(x*(2+5*x+x^2)/((1+x)*(1-x)^2), x,n+1),x,n),n=1..60); # Muniru A Asiru, Aug 06 2018

Select[Range[300],MemberQ[{2,7},Mod[#,8]]&] (* or *)
LinearRecurrence[ {1,1,-1},{2,7,10},60] (* Harvey P. Dale, Nov 05 2017 *)
CoefficientList[ Series[(x^2 + 5x + 2)/((x - 1)^2 (x + 1)), {x, 0, 60}], x] (* Robert G. Wilson v, Aug 07 2018 *)

makelist(4*n - mod(n,2) - 1, n, 1, 100); /* Franck Maminirina Ramaharo, Aug 06 2018 */

is(n) = #setintersect([n%8], [2, 7]) > 0 \\ Felix Fröhlich, Aug 06 2018

def A047524(n): return (n<<2)-1-(n&1) # Chai Wah Wu, Mar 30 2024

a(n) = 8*n - a(n-1) - 7, n > 1. - Vincenzo Librandi, Aug 06 2010
From R. J. Mathar, Mar 22 2011: (Start)
a(n) = 4*n - 3/2 + (-1)^n/2.
G.f.: x*(2+5*x+x^2) / ( (1+x)*(x-1)^2 ). (End)
From Franck Maminirina Ramaharo, Aug 06 2018: (Start)
a(n) = 4*n - (n mod 2) - 1.
a(n) = A047615(n) + 2.
a(2*n) = A004771(n-1).
a(2*n-1) = A017089(n-1).
E.g.f.: ((8*x - 3)*exp(x) + exp(-x) + 2)/2. (End)
a(n) = a(n-1) + a(n-2) - a(n-3). - Muniru A Asiru, Aug 06 2018
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)+2)*Pi/16 - log(2)/8 - sqrt(2)*log(sqrt(2)+1)/8. - Amiram Eldar, Dec 11 2021

More terms from Vincenzo Librandi, Aug 06 2010

A047535 Numbers that are congruent to {4, 7} mod 8.

Original entry on oeis.org

4, 7, 12, 15, 20, 23, 28, 31, 36, 39, 44, 47, 52, 55, 60, 63, 68, 71, 76, 79, 84, 87, 92, 95, 100, 103, 108, 111, 116, 119, 124, 127, 132, 135, 140, 143, 148, 151, 156, 159, 164, 167, 172, 175, 180, 183, 188, 191, 196, 199, 204, 207, 212, 215, 220, 223, 228, 231

Offset: 1

N. J. A. Sloane

Union of A004771 and A017113.

Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A004771, A017113, A047398, A047461, A047452, A047470, A047524, A047615, A047617.

A047535:=n->4*n - (1 + (-1)^n)/2; seq(A047535(n), n=1..100); # Wesley Ivan Hurt, Feb 24 2014

Table[4n - (1 + (-1)^n)/2, {n, 100}] (* Wesley Ivan Hurt, Feb 24 2014 *)

makelist(4*n - (1 + (-1)^n)/2, n, 1, 100); /* Franck Maminirina Ramaharo, Jul 22 2018 */

def A047535(n): return (n<<2)-(n&1^1) # Chai Wah Wu, Mar 30 2024

a(n) = 8*n - a(n-1) - 5 (with a(1)=4). - Vincenzo Librandi, Aug 06 2010
a(n) = 4*n -(1 + (-1)^n)/2. - Arkadiusz Wesolowski, Sep 18 2012
G.f.: x*(4+3*x+x^2) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Jul 10 2015
From Franck Maminirina Ramaharo, Jul 22 2018: (Start)
a(n) = A047470(n) + 4.
E.g.f.: (2 - exp(-x) + (8*x - 1)*exp(x))/2. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)+1)*Pi/16 - log(2)/4 - sqrt(2)*log(sqrt(2)+1)/8. - Amiram Eldar, Dec 11 2021

A047398 Numbers that are congruent to {3, 6} mod 8.

Original entry on oeis.org

3, 6, 11, 14, 19, 22, 27, 30, 35, 38, 43, 46, 51, 54, 59, 62, 67, 70, 75, 78, 83, 86, 91, 94, 99, 102, 107, 110, 115, 118, 123, 126, 131, 134, 139, 142, 147, 150, 155, 158, 163, 166, 171, 174, 179, 182, 187, 190, 195, 198, 203, 206, 211, 214, 219, 222, 227, 230

Offset: 1

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Union of A017101 and A017137.

Cf. A047461, A047452, A047470, A047524, A047535, A047615, A047617.

A047398:=n->4*n-(3+(-1)^n)/2: seq(A047398(n), n=1..100); # Wesley Ivan Hurt, Jan 30 2017

Flatten[# + {3, 6} & /@ (8 Range[0, 28])] (* Arkadiusz Wesolowski, Sep 25 2012 *)
LinearRecurrence[{1,1,-1},{3,6,11},60] (* Harvey P. Dale, Oct 26 2020 *)

makelist(4*n + mod(n, 2) - 2, n, 1, 100); /* Franck Maminirina Ramaharo, Aug 06 2018 */

def A047398(n): return ((n<<2)|(n&1))-2 # Chai Wah Wu, Mar 30 2024

a(n) = 8*n - a(n-1) - 7, n > 1. - Vincenzo Librandi, Aug 05 2010
From R. J. Mathar, Dec 05 2011: (Start)
a(n) = 4*n - (3 + (-1)^n)/2.
G.f.: x*(3+3*x+2*x^2) / ( (1+x)*(x-1)^2 ). (End)
From Franck Maminirina Ramaharo, Aug 06 2018: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3), n > 3.
a(n) = 4*n + (n mod 2) - 2.
a(n) = A047470(n) + 3.
a(2*n) = A017137(n-1).
a(2*n-1) = A017101(n-1).
E.g.f.: ((8*x - 3)*exp(x) - exp(-x) + 4)/2. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(2)*Pi/16 + log(2)/8 - sqrt(2)*log(sqrt(2)+1)/8. - Amiram Eldar, Dec 18 2021

A047615 Numbers that are congruent to {0, 5} mod 8.

Original entry on oeis.org

0, 5, 8, 13, 16, 21, 24, 29, 32, 37, 40, 45, 48, 53, 56, 61, 64, 69, 72, 77, 80, 85, 88, 93, 96, 101, 104, 109, 112, 117, 120, 125, 128, 133, 136, 141, 144, 149, 152, 157, 160, 165, 168, 173, 176, 181, 184, 189, 192, 197, 200, 205, 208, 213, 216, 221, 224, 229, 232

Offset: 1

N. J. A. Sloane

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Union of A008590 and A004770.

Cf. A047398, A047452, A047461, A047470, A047524, A047535, A047617.

Filtered([0..250], n->n mod 8=0 or n mod 8=5); # Muniru A Asiru, Jul 23 2018

[(8*n - 7 + (-1)^n)/2 : n in [1..50]]; // Wesley Ivan Hurt, Mar 26 2015

a:=n->add(4-(-1)^j, j=1..n): seq(a(n), n=0..59); # Zerinvary Lajos, Dec 13 2008

Table[(8 n - 7 + (-1)^n)/2, {n, 1, 40}] (* Wesley Ivan Hurt, Mar 26 2015 *)
Rest@ CoefficientList[Series[x^2*(5 + 3 x)/((1 - x)^2*(1 + x)), {x, 0, 59}], x] (* Michael De Vlieger, Aug 25 2016 *)
Rest@(Range[0, 60]! CoefficientList[ Series[(6 + Exp[-x] + (8 x - 7)*Exp[x])/2, {x, 0, 60}], x]) (* or *)
LinearRecurrence[{1, 1, -1}, {0, 5, 8}, 60] (* Robert G. Wilson v, Jul 23 2018 *)

forstep(n=0,200,[5,3],print1(n", ")) \\ Charles R Greathouse IV, Oct 17 2011

concat(0, Vec(x^2*(5+3*x)/((1-x)^2*(1+x)) + O(x^100))) \\ Colin Barker, Aug 25 2016

def A047615(n): return (n<<2)-3-(n&1) # Chai Wah Wu, Mar 30 2024

a(n) = 8*n-a(n-1)-11 (with a(1)=0). - Vincenzo Librandi, Aug 06 2010
a(n+1) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=5 and b(k)=2^(k+2) for k>0. - Philippe Deléham, Oct 17 2011
From Wesley Ivan Hurt, Mar 26 2015: (Start)
a(n) = a(n-1)+a(n-2)-a(n-3).
a(n) = (8n - 7 + (-1)^n)/2. (End)
G.f.: x^2*(5+3*x) / ((1-x)^2*(1+x)). - Colin Barker, Aug 25 2016
From Franck Maminirina Ramaharo, Jul 23 2018: (Start)
a(n) = A047470(n) - (-1)^(n - 1) + 1.
E.g.f.: (6 + exp(-x) + (8*x - 7)*exp(x))/2. (End)
Sum_{n>=2} (-1)^n/a(n) = log(2)/2 - (sqrt(2)-1)*Pi/16 - sqrt(2)*log(sqrt(2)+1)/8. - Amiram Eldar, Dec 18 2021

More terms from Vincenzo Librandi, Aug 06 2010

A376732 Triangle read by rows: T(n,k) is the maximum number of squares covered (i.e., attacked) by k independent (i.e., non-attacking) queens on an n X n chessboard.

Original entry on oeis.org

1, 4, 0, 9, 9, 0, 12, 15, 16, 16, 17, 23, 25, 25, 25, 20, 30, 35, 36, 36, 36, 25, 37, 45, 49, 49, 49, 49, 28, 44, 55, 62, 64, 64, 64, 64, 33, 52, 66, 76, 81, 81, 81, 81, 81, 36, 60, 77, 92, 100, 100, 100, 100, 100, 100, 41, 68, 88, 104, 121, 121, 121, 121, 121, 121, 121

Offset: 1

John King, Oct 03 2024

T(2,2) = T(3,3) = 0 indicate that there are no solutions to the n-queens problem when n is 2 or 3.

			Triangle begins:
  n\k|  1    2    3    4    5    6    7    8    9   10   11   12
 ----+-----------------------------------------------------------
   1 |  1;
   2 |  4,   0;
   3 |  9,   9,   0;
   4 | 12,  15,  16,  16;
   5 | 17,  23,  25,  25,  25;
   6 | 20,  30,  35,  36,  36,  36;
   7 | 25,  37,  45,  49,  49,  49,  49;
   8 | 28,  44,  55,  62,  64,  64,  64,  64;
   9 | 33,  52,  66,  76,  81,  81,  81,  81,  81;
  10 | 36,  60,  77,  92, 100, 100, 100, 100, 100, 100;
  11 | 41,  68,  88, 104, 121, 121, 121, 121, 121, 121, 121;
  12 | 44,  76, 101, 120, 134, 142, 144, 144, 144, 144, 144, 144;
  13 | 49,  84, 112, 136, 153, 165, 169, 169, 169, 169, 169, ...;
  14 | 52,  92, 125, 152, 172, 186, 194, 196, 196, 196, 196, ...;
  15 | 57, 100, 136, 168, 193, 209, 221, 224, 225, 225, 225, ...;
  16 | 60, 108, 149, 184, 212, 231, 242, 251, 256, 256, 256, ...;
  17 | 65, 116, 160, 200, 233, 255, 269, 281, 289, 289, 289, ...;
  18 | 68, 124, 173, 216, 252, 277, 294, 310, 322, 324, 324, ...;
  ...

Mia Muessig, Table of n, a(n) for n = 1..240
John King, Examples for Queens 1,2,3,4,5 up to 11x11.
John King, Examples for Queens 6,7,8,9 up to 15x15.
Mia Muessig, Julia code to compute the sequence

Columns 1..8 are A047461, A374933, A375116, A374934, A374935, A374936, A374937, A374938.

Cf. A075324, A002567, A075458, A274947.

T(n,k) = n^2 for k >= A075324(n), n >= 4.

Initial terms by John King and Mia Müßig added by Mia Muessig, Oct 05 2024

A374933 Maximum number of squares covered (i.e., attacked) by 2 independent (i.e., non-attacking) queens on an n X n chessboard.

Original entry on oeis.org

9, 15, 23, 30, 37, 44, 52, 60, 68, 76, 84, 92, 100, 108, 116, 124, 132, 140, 148, 156, 164, 172, 180, 188, 196, 204, 212, 220, 228, 236, 244, 252, 260, 268, 276, 284, 292, 300, 308, 316, 324, 332, 340, 348, 356, 364, 372, 380, 388, 396, 404, 412, 420

Offset: 3

John King, Jul 24 2024

It is not possible to place two non-attacking queens on a 1 X 1 or 2 X 2 chessboard.

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A017113, A047461 (case for one queen).

a(n) = 8*n - 20 for n >= 8.

G.f.: x^3*(9 - 3*x + 2*x^2 - x^3 + x^6)/(1 - x)^2. - Stefano Spezia, Jul 25 2024

A374934 Maximum number of squares covered (i.e., attacked) by 4 independent (i.e., nonattacking) queens on an n X n chessboard.

Original entry on oeis.org

16, 25, 36, 49, 62, 76, 92, 104, 120, 136, 152, 168, 184, 200, 216

Offset: 4

John King, Aug 08 2024

			4 X 4:
  x Q x x
  x x x Q
  Q x x x
  x x Q x
5 X 5 there are several arrangements:
  x Q x x x
  x x x x x
  x x x x Q
  Q x x x x
  x x x Q x
6 X 6 and 7 X 7 (add a row and column) pattern as 4 queens knight-1,3 and 1,4 separation (not symmetric):
  . . . . . . .
  x x x x Q x .
  Q x x x x x .
  x x x x x x .
  x x x x x Q .
  x Q x x x x .
8 X 8: queens all knight-1,4 apart;
8 X 8 has 2 o/s;
9 X 9 has 5 o/s;
10 X 10 has 8 o/s;
  o x x x x x x x x o
  x o x x x x x x o x
  x x x Q x x x x x x
  x x x x x x x Q x x
  x x x x x x x x x x
  x x x x x x x x x x
  x x Q x x x x x x x
  x x x x x x Q x x x
  x o x x x x x x o x
  o x x x x x x x x o
beyond 10 X 10, the 4 queens separated as 1,2 knights begins to be the best layout; at 15 X 15, the pattern is clear.
  o x x o o x x x x o o x x o x
  x o x x o x x x x o x x o x x
  x x o x x x x x x x x o x x o
  o x x o x x x x x x o x x o o
  o o x x o x x x x o x x o o o
  x x x x x x Q x x x x x x x x
  x x x x x x x x Q x x x x x x
  x x x x x Q x x x x x x x x x
  x x x x x x x Q x x x x x x x
  o o x x o x x x x o x x o o o
  o x x o x x x x x x o x x o o
  x x o x x x x x x x x o x x o
  x o x x o x x x x o x x o x x
  o x x o o x x x x o o x x o x
  x x o o o x x x x o o o x x o

Column 4 of A376732.

Cf. A017113, A047461, A374933, A375116, A374935, A374936, A374937, A374938.

a(18) added using data from Mia Muessig by Andrew Howroyd, Oct 05 2024

cp's OEIS Frontend

A047470 Numbers that are congruent to {0, 3} mod 8.

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A047452 Numbers that are congruent to {1, 6} mod 8.

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A047617 Numbers that are congruent to {2, 5} mod 8.

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A047524 Numbers that are congruent to {2, 7} mod 8.

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A047535 Numbers that are congruent to {4, 7} mod 8.

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A047398 Numbers that are congruent to {3, 6} mod 8.

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