A047461 -id:A047461 - cp's OEIS frontend

A374935 Maximum number of squares covered (i.e., attacked) by 5 independent (i.e., nonattacking) queens on an n X n chessboard.

25, 36, 49, 64, 81, 100, 121, 134, 153, 172, 193, 212, 233, 252

Offset: 5

John King, Aug 08 2024

			5 X 5; 6 X 6; 7 X 7; 8 X 8;  Center-square +4Queens separated as if 1,2 knights.
              at 11 X 11 and beyond this pattern seems to be 'best'.
  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 Q x x
  x x x Q x x x x
  x Q x x x x x x
  x x x x Q x x x
  x x x x x x x x
9 X 9; 10 X 10; 11 X 11; Center-square +4Queens separated as 2,4 knights.
  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 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 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 x Q 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 x x x x x

Column 5 of A376732.

Cf. A075324, A047461, A374933, A375116, A374933, A374934, A374936.

Unverified a(19) removed by Andrew Howroyd, Oct 05 2024

A374936 Maximum number of squares covered (i.e., attacked) by 6 independent (i.e., nonattacking) queens on an n X n chessboard.

Original entry on oeis.org

36, 49, 64, 81, 100, 121, 142, 165, 186, 209, 231, 255, 277

Offset: 6

John King, Aug 08 2024

			Example for 12 X 12: There are 2 cells marked 'o' or uncovered thus a(12) = 12 * 12 - 2 = 142.
  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
  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 Q 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 x
  o x x x x x x x x x x x
  x x x x x x x x x x x x
  x x x x x x x x x x x x
  x x x x x x x x o x x x
From _Christian Sievers_, Sep 08 2024: (Start)
Example for 14 X 14 with 186 attacked squares (unattacked ones marked with "+"):
  . . Q . . . . . . . . . . .
  . . . . . . . . . Q . . . .
  . . . . . . . . . . . . . +
  . + . . . . . . . . . . . .
  . . . Q . . . . . . . . . .
  . . . . . . . . . . . . . .
  . . . . . . . . . . . . . .
  . . . . . . . . . . . . Q .
  . + . . . . . . . . . . . .
  . . . . . . . . . . . . . +
  . . . . . . Q . . . . . . .
  . + . . + . . . . . . + . .
  . . . . . + . . . . + . . +
  Q . . . . . . . . . . . . .
(End)

Column 6 of A376732.

Cf. A075324, A047461, A374933, A375116, A374933, A374934, A374935, A374937, A374938.

a(14) corrected and a(15) confirmed by Christian Sievers, Sep 08 2024

a(16)-a(18) added using data from Mia Muessig by Andrew Howroyd, Oct 05 2024

A375116 Maximum number of squares covered (i.e., attacked) by 3 independent (i.e., nonattacking) queens on an n X n chessboard.

Original entry on oeis.org

16, 25, 35, 45, 55, 66, 77, 88, 101, 112, 125, 136, 149, 160, 173, 184, 197, 208, 221, 232, 245, 256, 269, 280, 293, 304, 317, 328, 341, 352, 365, 376, 389, 400, 413, 424, 437, 448, 461, 472, 485, 496, 509, 520, 533, 544, 557, 568, 581, 592, 605, 616, 629, 640, 653, 664, 677

Offset: 4

John King, Jul 30 2024

nonn

It is not possible to place 3 independent queens on a 1 X 1 or 2 X 2 or 3 X 3 board.
There is a related sequence of 'uncovered' squares i.e., n^2 - a(n).
There is another sequence denoting the potency of the new queen a(n) - A374933(n).

			4 X 4 complete coverage with 3 queens
  x x x x
  x Q x x
  x x x Q
  Q x x x
5 X 5 complete coverage with 3 queens
  Q x x x x
  x x x x x
  x x x Q x
  x x x x x
  x x Q x x
6 X 6 incomplete 1 o/s
  x x x x o x
  Q x x x x x
  x x x x x Q
  x x x x x x
  x x Q x x x
  x x x x x x
6 X 6 coverage complete but NOT independent
  Q 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 q x x x
  x x x x x x
7 X 7 best leaves 4 o/s  (same layout as 6 X 6 with extra row and column)
There are alternative layouts - how many is not identified.
  x x x x o 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 Q x x x x
  x x x x x x o
  x x x o x x o

Column 3 of A376732.

Cf. A047461 (for one queen), A374933 (for two queens), A374934, A374935, A374936.

a(n) = 12*n - 43 - (n mod 2) for n >= 10.

a(6)-a(8) corrected by John King, Sep 17 2024

a(9) corrected using data from Mia Muessig by Andrew Howroyd, Oct 05 2024

A047415 Numbers that are congruent to {1, 3, 4, 6} mod 8.

Original entry on oeis.org

1, 3, 4, 6, 9, 11, 12, 14, 17, 19, 20, 22, 25, 27, 28, 30, 33, 35, 36, 38, 41, 43, 44, 46, 49, 51, 52, 54, 57, 59, 60, 62, 65, 67, 68, 70, 73, 75, 76, 78, 81, 83, 84, 86, 89, 91, 92, 94, 97, 99, 100, 102, 105, 107, 108, 110, 113, 115, 116, 118, 121, 123, 124

Offset: 1

N. J. A. Sloane

Consider an operation SS(n) defined for a specific sequence b where b(n) is the n-th term of b. This operation is defined as follows: SS(1) = b(1); if b(n+1) > SS(n), SS(n+1) = SS(n) + b(n+1), otherwise SS(n+1) = SS(n) - b(n+1) (If b(n) = A000027(n), then SS(n) = A008344(n+1)). If the sequence b can represent any permutation of the first n natural numbers, then a(n) is the maximum possible value of SS(n). - Iain Fox, Sep 15 2020 (see link by Math StackExchage)

Iain Fox, Table of n, a(n) for n = 1..10000
Math StackExchange, Strange Sum of Numbers 1 to 100, September 2020.
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Cf. A008344, A047398, A047461.

[n : n in [0..150] | n mod 8 in [1, 3, 4, 6]]; // Wesley Ivan Hurt, May 31 2016

A047415:=n->2*(n-1)-(I^(n*(n+1))-1)/2: seq(A047415(n), n=1..100); # Wesley Ivan Hurt, May 31 2016

Select[Range[108], MemberQ[{1, 3, 4, 6}, Mod[#, 8]]&] (* Bruno Berselli, Dec 06 2011 *)

makelist(2*(n-1)-(%i^(n*(n+1))-1)/2, n, 1, 55); /* Bruno Berselli, Dec 06 2011 */

a(n)=2*(n-1)-(I^(n*(n+1))-1)/2 \\ Charles R Greathouse IV, Dec 06 2011

G.f.: x*(1+x+2*x^3) / ( (1+x^2)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011
a(n) = 2*(n-1)-(i^(n*(n+1))-1)/2, where i=sqrt(-1). - Bruno Berselli, Dec 06 2011
From Wesley Ivan Hurt, May 31 2016: (Start)
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4.
a(n) = (1+i)*(4*n-4*n*i+3*i-3+i^(1-n)-i^n)/4 where i=sqrt(-1).
a(2*k) = A047398(k), a(2*k-1) = A047461(k). (End)
E.g.f.: (4 + sin(x) - cos(x) + (4*x - 3)*exp(x))/2. - Ilya Gutkovskiy, May 31 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*Pi/16 - (sqrt(2)+1)*log(2)/8 + sqrt(2)*log(sqrt(2)+2)/4. - Amiram Eldar, Dec 24 2021

A153125 Triangle read by rows: T(n,k) = maximal number of squares that can be covered by a queen on an n X k chessboard, 1<=k<=n.

Original entry on oeis.org

1, 2, 4, 3, 6, 9, 4, 7, 10, 12, 5, 8, 11, 14, 17, 6, 9, 12, 15, 18, 20, 7, 10, 13, 16, 19, 22, 25, 8, 11, 14, 17, 20, 23, 26, 28, 9, 12, 15, 18, 21, 24, 27, 30, 33, 10, 13, 16, 19, 22, 25, 28, 31, 34, 36, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, 12, 15, 18, 21, 24, 27, 30, 33, 36

Offset: 1

Reinhard Zumkeller, Dec 20 2008

Sums of rows give A153126; central terms give A016861;
A047461(n) = T(n,n);
T(n,2*k-1) = T(n-1,2*k-1) + 1 for 2*k-1

			Triangle T(n,k) begins:
1;
2,  4;
3,  6,  9;
4,  7, 10, 12;
5,  8, 11, 14, 17;
6,  9, 12, 15, 18, 20;
7, 10, 13, 16, 19, 22, 25;
8, 11, 14, 17, 20, 23, 26, 28;

T[n_,k_]:=n+3*(k-1)-(1-Mod[n,2])*If[k==n,1,0];
Flatten[Table[Table[T[n,k],{k,1,n}],{n,1,20}]]
(* From Vaclav Kotesovec, Sep 07 2012 *)

T(n,k) = n + 3*(k-1) - (1 - n Mod 2)*delta_{n,k}, 1<=k<=n; delta is the Kronecker symbol.

A279406 Irregular triangle read by rows: T(n,k) (n>=1, 0 <= k <= n^2) = minimal number of squares attacked by k queens on an n X n toroidal board.

Original entry on oeis.org

0, 1, 0, 4, 4, 4, 4, 0, 9, 9, 9, 9, 9, 9, 9, 9, 9, 0, 12, 14, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 0, 17, 22, 23, 24, 24, 24, 24, 24, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 0, 20, 26, 30, 30, 32, 32, 34, 34, 34

Offset: 1

Andrey Zabolotskiy, Dec 11 2016

A279405(n) is maximal m such that T(n,m) >= m.

Generally, T(n,k') <= n^2-k if and only if T(n,k) <= n^2-k'.

			The triangle begins:
0 1
0 4 4 4 4
0 9 9 9 9 9 9 9 9 9
0 12 14 15 15 16 16 16 16 16 16 16 16 16 16 16 16

Andrey Zabolotskiy, First 9 rows of the triangle and a part of row 10

Cf. A000290, A250000, A274947, A274948, A279405, A279403.

T(n,0) = 0.
T(n,k) = A000290(n) for k > A000290(n) - T(n,1).
T(n,1) = A047461(n) = A000290(n) - A279403(n,1).

A047466 Numbers that are congruent to {0, 1, 2, 4} mod 8.

Original entry on oeis.org

0, 1, 2, 4, 8, 9, 10, 12, 16, 17, 18, 20, 24, 25, 26, 28, 32, 33, 34, 36, 40, 41, 42, 44, 48, 49, 50, 52, 56, 57, 58, 60, 64, 65, 66, 68, 72, 73, 74, 76, 80, 81, 82, 84, 88, 89, 90, 92, 96, 97, 98, 100, 104, 105, 106, 108, 112, 113, 114, 116, 120, 121, 122

Offset: 1

N. J. A. Sloane

Bruno Berselli, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Essentially the same as A003485.

Cf. A047461, A047467.

[n: n in [0..120] | n mod 8 in [0,1,2,4]];

A047466:=n->2*n-4+(3-I^(2*n))*(1-I^(n*(n+1)))/4: seq(A047466(n), n=1..100); # Wesley Ivan Hurt, Jun 01 2016

Select[Range[0,120], MemberQ[{0, 1, 2, 4}, Mod[#, 8]] &] (* or *) LinearRecurrence[{1, 0, 0, 1, -1}, {0, 1, 2, 4, 8}, 60] (* Bruno Berselli, Jul 18 2012 *)

makelist(2*n-4+(3-(-1)^n)*(1-%i^(n*(n+1)))/4,n,1,60);

concat(0, Vec((1+x+2*x^2+4*x^3)/((1+x)*(1+x^2)*(1-x)^2)+O(x^60))) (End)

G.f.: x^2*(1+x+2*x^2+4*x^3) / ( (1+x)*(1+x^2)*(1-x)^2 ). - R. J. Mathar, Oct 08 2011
a(n) = 2*n-4+(3-(-1)^n)*(1-i^(n*(n+1)))/4, where i=sqrt(-1). - Bruno Berselli, Jul 18 2012
From Wesley Ivan Hurt, Jun 01 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(2k) = A047461(k), a(2k-1) = A047467(k). (End)
Sum_{n>=2} (-1)^n/a(n) = (1+2*sqrt(2))*Pi/32 + (3+sqrt(2))*log(2)/16 - sqrt(2)*log(2-sqrt(2))/8. - Amiram Eldar, Dec 20 2021

A047541 Numbers that are congruent to {1, 2, 4, 7} mod 8.

Original entry on oeis.org

1, 2, 4, 7, 9, 10, 12, 15, 17, 18, 20, 23, 25, 26, 28, 31, 33, 34, 36, 39, 41, 42, 44, 47, 49, 50, 52, 55, 57, 58, 60, 63, 65, 66, 68, 71, 73, 74, 76, 79, 81, 82, 84, 87, 89, 90, 92, 95, 97, 98, 100, 103, 105, 106, 108, 111, 113, 114, 116, 119, 121, 122, 124

Offset: 1

N. J. A. Sloane

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

Cf. A047461, A047524.

[n : n in [0..150] | n mod 8 in [1, 2, 4, 7]]; // Wesley Ivan Hurt, Jun 04 2016

A047541:=n->(1+I)*(n*(4-4*I)+3*I-3+I^(-n)-I^(1+n))/4: seq(A047541(n), n=1..100); # Wesley Ivan Hurt, Jun 04 2016

Table[(1+I)*(n*(4-4*I)+3*I-3+I^(-n)-I^(1+n))/4, {n, 80}] (* Wesley Ivan Hurt, Jun 04 2016 *)
Select[Range[200],MemberQ[{1,2,4,7},Mod[#,8]]&] (* or  *) LinearRecurrence[ {2,-2,2,-1},{1,2,4,7},70] (* Harvey P. Dale, Jul 09 2020 *)

a(n)=n\4*8+[-1,1,2,4][n%4+1] \\ Charles R Greathouse IV, Nov 04 2011

From Wesley Ivan Hurt, Jun 04 2016: (Start)
G.f.: x*(1+2*x^2+x^3)/(x-1)^2*(1+x^2).
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4.
a(n) = (1+i)*(n*(4-4*i)+3*i-3+i^(-n)-i^(1+n))/4 where i=sqrt(-1).
a(2k) = A047524(k), a(2k-1) = A047461(k). (End)
E.g.f.: (2 + sin(x) + cos(x) + (4*x - 3)*exp(x))/2. - Ilya Gutkovskiy, Jun 04 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = (2*sqrt(2)+1)*Pi/16 - log(2)/8. - Amiram Eldar, Dec 24 2021

A047613 Numbers that are congruent to {1, 2, 4, 5} mod 8.

Original entry on oeis.org

1, 2, 4, 5, 9, 10, 12, 13, 17, 18, 20, 21, 25, 26, 28, 29, 33, 34, 36, 37, 41, 42, 44, 45, 49, 50, 52, 53, 57, 58, 60, 61, 65, 66, 68, 69, 73, 74, 76, 77, 81, 82, 84, 85, 89, 90, 92, 93, 97, 98, 100, 101, 105, 106, 108, 109, 113, 114, 116, 117, 121, 122, 124

Offset: 1

N. J. A. Sloane

Bruno Berselli, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A047461, A047617.

[n: n in [1..120] | n mod 8 in [1,2,4,5]];

A047613:=n->2*n-2-(I^(2*n)+I^(n*(n+1)))/2: seq(A047613(n), n=1..100); # Wesley Ivan Hurt, Jun 02 2016

Select[Range[120], MemberQ[{1, 2, 4, 5}, Mod[#, 8]] &] (* or *) LinearRecurrence[{1, 0, 0, 1, -1}, {1, 2, 4, 5, 9}, 60] (* Bruno Berselli, Jul 17 2012 *)

makelist(2*n-2-((-1)^n+%i^(n*(n+1)))/2,n,1,60);

Vec((1+x+2*x^2+x^3+3*x^4)/((1+x)*(1-x)^2*(1+x^2))+O(x^60)) (End)

From Bruno Berselli, Jul 17 2012: (Start)
G.f.: x*(1+x+2*x^2+x^3+3*x^4)/((1+x)*(1-x)^2*(1+x^2)).
a(n) = 2*n-2-((-1)^n+i^(n*(n+1)))/2, where i=sqrt(-1). (End)
From Wesley Ivan Hurt, Jun 02 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(2k) = A047617(k), a(2k-1) = A047461(k). (End)
E.g.f.: (6 + sin(x) - cos(x) + (4*x - 3)*sinh(x) + (4*x - 5)*cosh(x))/2. - Ilya Gutkovskiy, Jun 02 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = (2*sqrt(2)-1)*Pi/16 + sqrt(2)*log(sqrt(2)+2)/4 - (sqrt(2)+1)*log(2)/8. - Amiram Eldar, Dec 23 2021

A128622 Triangle T(n, k) = A128064(unsigned) * A128174, read by rows.

Original entry on oeis.org

1, 1, 2, 3, 2, 3, 3, 4, 3, 4, 5, 4, 5, 4, 5, 5, 6, 5, 6, 5, 6, 7, 6, 7, 6, 7, 6, 7, 7, 8, 7, 8, 7, 8, 7, 8, 9, 8, 9, 8, 9, 8, 9, 8, 9, 9, 10, 9, 10, 9, 10, 9, 10, 9, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 11, 12, 11, 12, 11, 12, 11, 12, 11, 12, 11, 12

Offset: 1

Gary W. Adamson, Mar 14 2007

			First few rows of the triangle are:
  1;
  1, 2;
  3, 2, 3;
  3, 4, 3, 4;
  5, 4, 5, 4, 5;
  5, 6, 5, 6, 5, 6;
  7, 6, 7, 6, 7, 6, 7;
  ...

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Cf. A000027, A016777, A042963, A047461, A109613, A128064, A128174.

Cf. A000326 (diagonal sums), A014848 (row sums), A319556 (alternating row sums).

[n - ((n+k) mod 2): k in [1..n], n in [1..16]]; // G. C. Greubel, Mar 14 2024

Table[n - Mod[n+k,2], {n,16}, {k,n}]//Flatten (* G. C. Greubel, Mar 14 2024 *)

flatten([[n - ((n+k)%2) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Mar 14 2024

T(n, k) = abs(A128064(n,k) * A128174(n, k), as infinite lower triangular matrices.
Sum_{k=1..n} T(n, k) = A014848(n) (row sums).
From G. C. Greubel, Mar 14 2024: (Start)
T(n, k) = n - (1 - (-1)^(n+k))/2 = n - (n+k mod 2).
T(n, 1) = A109613(n+1).
T(n, n) = A000027(n).
T(2*n-1, n) = A042963(n).
T(3*n-1, n) = A016777(n+1).
T(4*n-3, n) = A047461(n).
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = A319556(n).
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = A000326(floor((n+1)/2)).
Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1, k) = A123684(floor((n+1)/2)). (End)

More terms added by G. C. Greubel, Mar 14 2024

Previous Showing 11-20 of 27 results. Next

cp's OEIS Frontend

A374935 Maximum number of squares covered (i.e., attacked) by 5 independent (i.e., nonattacking) queens on an n X n chessboard.

Views

Author

Keywords

Examples

Crossrefs

Extensions

A374936 Maximum number of squares covered (i.e., attacked) by 6 independent (i.e., nonattacking) queens on an n X n chessboard.

Views

Author

Keywords

Examples

Crossrefs

Extensions

A375116 Maximum number of squares covered (i.e., attacked) by 3 independent (i.e., nonattacking) queens on an n X n chessboard.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A047415 Numbers that are congruent to {1, 3, 4, 6} mod 8.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Maxima

PARI

Formula

A153125 Triangle read by rows: T(n,k) = maximal number of squares that can be covered by a queen on an n X k chessboard, 1<=k<=n.

Views

Author

Keywords

Comments

Examples

Programs

Mathematica

Formula

A279406 Irregular triangle read by rows: T(n,k) (n>=1, 0 <= k <= n^2) = minimal number of squares attacked by k queens on an n X n toroidal board.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A047466 Numbers that are congruent to {0, 1, 2, 4} mod 8.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Maxima

PARI

Formula

A047541 Numbers that are congruent to {1, 2, 4, 7} mod 8.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI