A047461 -id:A047461 - cp's OEIS frontend

A279403 Irregular triangle read by rows: T(n,k) (n>=1, 0 <= k <= n^2) = minimal number of squares not attacked by k queens on an n X n toroidal board, with trailing zeros truncated.

1, 4, 9, 16, 4, 25, 8, 2, 36, 16, 4, 49, 24, 11, 4, 64, 36, 16, 6, 81, 48, 27, 12, 3, 100, 64, 36, 19, 4, 121, 80, 51, 29, 13, 144, 100, 64, 39, 16, 6, 169, 120, 83, 53, 29, 8, 2, 196, 144, 100, 67, 36, 18, 8, 225, 168, 223, 82, 41, 256, 196, 144, 103, 64, 40

Offset: 1

Andrey Zabolotskiy, Dec 11 2016

Row lengths are A279402.

			The triangle begins:
1 (0)
4 (0, 0, 0, 0)
9 (0, 0, ...)
16 4 (0, 0, ...)
25 8 2
36 16 4
49 24 11 4
64 36 16 6
81 48 27 12 3
100 64 36 19 4
121 80 51 29 13
144 100 64 39 16 6
169 120 83 53 29 8 2
196 144 100 67 36 18 8
225 168 123 82 41
256 196 144 103 64 40 ...

Cf. A279402, A279406.

T(n,0) = A000290(n).
T(n,1) = A000290(n)-A047461(n) = A137932(n-1).
T(n,2) = A248825(n-4) for n >= 6.

A047411 Numbers that are congruent to {1, 2, 4, 6} mod 8.

Original entry on oeis.org

1, 2, 4, 6, 9, 10, 12, 14, 17, 18, 20, 22, 25, 26, 28, 30, 33, 34, 36, 38, 41, 42, 44, 46, 49, 50, 52, 54, 57, 58, 60, 62, 65, 66, 68, 70, 73, 74, 76, 78, 81, 82, 84, 86, 89, 90, 92, 94, 97, 98, 100, 102, 105, 106, 108, 110, 113, 114, 116, 118, 121, 122, 124, 126, 129, 130

Offset: 1

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A016825, A047461.

[n : n in [0..150] | n mod 8 in [1, 2, 4, 6]]; // Wesley Ivan Hurt, May 22 2016

A047411 := proc(n) if n <= 4 then op(n,[1,2,4,6]); else procname(n-4)+8; end if; end proc: seq(A047411(n), n=1..99); # R. J. Mathar, Feb 11 2010

Table[(8n-7-I^(2n)+I^(1-n)-I^(1+n))/4, {n, 80}] (* Wesley Ivan Hurt, May 22 2016 *)
LinearRecurrence[{1,0,0,1,-1}, {1,2,4,6,9}, 50] (* G. C. Greubel, May 23 2016 *)

From R. J. Mathar, Mar 10 2008: (Start)
a(n) = a(n-4)+8.
O.g.f.: 2/(-1+x)^2+1/(2*(x^2+1))+7/(4*(-1+x))+1/(4*(x+1)). (End)
a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5. - R. J. Mathar, Feb 11 2010
From Wesley Ivan Hurt, May 22 2016: (Start)
a(n) = (8*n-7-i^(2*n)+i^(1-n)-i^(1+n))/4 where i=sqrt(-1).
a(2*n+2) = A016825(n) n>0, a(2*n-1) = A047461(n). (End)
E.g.f.: (4 + sin(x) + (4*x - 3)*sinh(x) + 4*(x - 1)*cosh(x))/2. - Ilya Gutkovskiy, May 23 2016
a(n) = (8*n-7-cos(n*Pi)+2*sin(n*Pi/2))/4. - Wesley Ivan Hurt, Oct 05 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)+1)*Pi/16 - sqrt(2)*log(3-2*sqrt(2))/16. - Amiram Eldar, Dec 23 2021

Extended by R. J. Mathar, Feb 11 2010

A047460 Numbers that are congruent to {0, 1, 3, 4} mod 8.

Original entry on oeis.org

0, 1, 3, 4, 8, 9, 11, 12, 16, 17, 19, 20, 24, 25, 27, 28, 32, 33, 35, 36, 40, 41, 43, 44, 48, 49, 51, 52, 56, 57, 59, 60, 64, 65, 67, 68, 72, 73, 75, 76, 80, 81, 83, 84, 88, 89, 91, 92, 96, 97, 99, 100, 104, 105, 107, 108, 112, 113, 115, 116, 120, 121, 123

Offset: 1

N. J. A. Sloane

Vincenzo Librandi, 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, A047470.

I:=[0, 1, 3, 4, 8]; [n le 5 select I[n] else Self(n-1)+Self(n-4)-Self(n-5): n in [1..70]]; // Vincenzo Librandi, May 16 2012

A047460:=n->(-1/4+I/4)*((6+6*I)+(1+I)*I^(2*n)+(-I)^n+I*I^n)+2*n: seq(A047460(n), n=1..100); # Wesley Ivan Hurt, Jun 01 2016

Select[Range[0,3000], MemberQ[{0,1,3,4}, Mod[#,8]]&] (* Vincenzo Librandi, May 16 2012 *)

my(x='x+O('x^100)); concat(0, Vec(x^2*(1+2*x+x^2+4*x^3)/((1-x)^2*(1+x)*(1+x^2)))) \\ Altug Alkan, Dec 24 2015

From Colin Barker, May 14 2012: (Start)
a(n) = (-1/4+i/4)*((6+6*i)+(1+i)*(-1)^n+(-i)^n+i*i^n)+2*n where i=sqrt(-1).
G.f.: x^2*(1+2*x+x^2+4*x^3)/((1-x)^2*(1+x)*(1+x^2)). (End)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5. - Vincenzo Librandi, May 16 2012
a(2k) = A047461(k), a(2k-1) = A047470(k). - Wesley Ivan Hurt, Jun 01 2016
Sum_{n>=2} (-1)^n/a(n) = Pi/8 + (2-sqrt(2))*log(2)/8 + sqrt(2)*log(2+sqrt(2))/4. - Amiram Eldar, Dec 20 2021

A047544 Numbers that are congruent to {1, 3, 4, 7} mod 8.

Original entry on oeis.org

1, 3, 4, 7, 9, 11, 12, 15, 17, 19, 20, 23, 25, 27, 28, 31, 33, 35, 36, 39, 41, 43, 44, 47, 49, 51, 52, 55, 57, 59, 60, 63, 65, 67, 68, 71, 73, 75, 76, 79, 81, 83, 84, 87, 89, 91, 92, 95, 97, 99, 100, 103, 105, 107, 108, 111, 113, 115, 116, 119, 121, 123, 124

Offset: 1

N. J. A. Sloane

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

Cf. A004767, A047461.

[n : n in [0..150] | n mod 8 in [1, 3, 4, 7]]; // Wesley Ivan Hurt, May 29 2016

A047544:=n->(8*n-5+I^(2*n)+I^(1-n)-I^(1+n))/4: seq(A047544(n), n=1..100); # Wesley Ivan Hurt, May 29 2016

Table[(8n-5+I^(2n)+I^(1-n)-I^(1+n))/4, {n, 80}] (* Wesley Ivan Hurt, May 29 2016 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {1, 3, 4, 7, 9}, 50] (* G. C. Greubel, May 29 2016 *)

From Wesley Ivan Hurt, May 29 2016: (Start)
G.f.: x*(1+2*x+x^2+3*x^3+x^4) / ((x-1)^2*(1+x+x^2+x^3)).
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (8*n-5+i^(2*n)+i^(1-n)-i^(1+n))/4 where i=sqrt(-1).
a(2k) = A004767(k-1) for n>0, a(2k-1) = A047461(k). (End)
E.g.f.: (2 + sin(x) + (4*x - 3)*sinh(x) + (4*x - 2)*cosh(x))/2. - Ilya Gutkovskiy, May 29 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)+3)*Pi/16 - log(2)/4 + sqrt(2)*log(sqrt(2)+1)/8. - Amiram Eldar, Dec 24 2021

A047600 Numbers that are congruent to {1, 3, 4, 5} mod 8.

Original entry on oeis.org

1, 3, 4, 5, 9, 11, 12, 13, 17, 19, 20, 21, 25, 27, 28, 29, 33, 35, 36, 37, 41, 43, 44, 45, 49, 51, 52, 53, 57, 59, 60, 61, 65, 67, 68, 69, 73, 75, 76, 77, 81, 83, 84, 85, 89, 91, 92, 93, 97, 99, 100, 101, 105, 107, 108, 109, 113, 115, 116, 117, 121, 123, 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, A047621.

[n: n in [1..120] | n mod 8 in [1,3,4,5]];

A047600:=n->2*n-1-(3+I^(2*n))*(1+I^(n*(n+1)))/4: seq(A047600(n), n=1..100); # Wesley Ivan Hurt, Jun 02 2016

Select[Range[120], MemberQ[{1,3,4,5}, Mod[#,8]]&]  (* Harvey P. Dale, Mar 09 2011 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {1, 3, 4, 5, 9}, 60] (* Bruno Berselli, Jul 17 2012 *)

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

Vec((1+2*x+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+2*x+x^2+x^3+3*x^4)/((1+x)*(1-x)^2*(1+x^2)).
a(n) = 2*n-1 -(3+(-1)^n)*(1+i^(n*(n+1)))/4, 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) = A047621(k), a(2k-1) = A047461(k). (End)
E.g.f.: (6 + sin(x) - 2*cos(x) + (4*x - 3)*sinh(x) + 4*(x - 1)*cosh(x))/2. - Ilya Gutkovskiy, Jun 03 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)+1)*Pi/16 - (4+3*sqrt(2))*log(2)/16 + 3*sqrt(2)*log(sqrt(2)+2)/8. - Amiram Eldar, Dec 24 2021

A369801 Maximum number of segments between n points on a circle so that they can be colored in 2 colors so that each of them intersects (at an internal point) at most one other segment of the same color.

Original entry on oeis.org

1, 3, 6, 10, 15, 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

Offset: 2

Mladen Valkov, Feb 01 2024

Bulgarian Winter Mathematical Competition "Atanas Radev", Problems and solutions brochure, Problem 9.4, p. 6 (in Bulgarian).
Art of Problem Solving, High School Olympiads - Colored segments, 2024.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A047452, A047461.

Drop[CoefficientList[Series[ -x^2*(x^7-2*x^4-2*x^3-2*x^2-2*x-1)/((x+1)*(x-1)^2),{x,0,59}],x],2] (* James C. McMahon, Mar 08 2024 *)

def A369801(n): return (n-2<<2)-(n&1) if n>=7 else (1, 3, 6, 10, 15)[n-2] # Chai Wah Wu, Mar 30 2024

a(n) = n*(n-1)/2 for n<=6, a(2*k+1) = 8*k-5 if k>=3, a(2*k) = 8*k-8 if k>=4.
G.f.: -x^2*(x^7-2*x^4-2*x^3-2*x^2-2*x-1)/((x+1)*(x-1)^2).
a(n) = A047452(n-2) + 2 = A047461(n-1) - 1 for n >= 7. - Hugo Pfoertner, Feb 05 2024

A109534 a(0)=1, a(n)=n+a(n-1) if n mod 2=0, a(n)=3n-a(n-1) if n mod 2 = 1.

Original entry on oeis.org

1, 2, 4, 5, 9, 6, 12, 9, 17, 10, 20, 13, 25, 14, 28, 17, 33, 18, 36, 21, 41, 22, 44, 25, 49, 26, 52, 29, 57, 30, 60, 33, 65, 34, 68, 37, 73, 38, 76, 41, 81, 42, 84, 45, 89, 46, 92, 49, 97, 50, 100, 53, 105, 54, 108, 57, 113, 58, 116, 61, 121, 62, 124, 65, 129, 66, 132, 69, 137

Offset: 0

Roger L. Bagula, Jun 18 2005

nonn

Index entries for linear recurrences with constant coefficients, signature (0, 1, 0, 1, 0, -1).

a:=proc(n) if n=0 then 1 elif n mod 2=0 then n+a(n-1) else 3*n-a(n-1) fi end: seq(a(n),n=0..68);

a[0] = 1; a[n_] := a[n] = If[Mod[n, 2] == 0, a[n - 1] + n, -a[n - 1] + 3*n] aa = Table[a[n], {n, 0, 100}]

G.f.: (1+2x+3x^2+3x^3+4x^4-x^5)/((1+x^2)(1-x)^2(1+x)^2). a(n)=a(n-2)+a(n-4)-a(n-6). a(2n)=A047461(n+1). a(2n+1)=A042963(n+1). [From R. J. Mathar, Oct 29 2008]

cp's OEIS Frontend

A279403 Irregular triangle read by rows: T(n,k) (n>=1, 0 <= k <= n^2) = minimal number of squares not attacked by k queens on an n X n toroidal board, with trailing zeros truncated.

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A047411 Numbers that are congruent to {1, 2, 4, 6} mod 8.

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A047460 Numbers that are congruent to {0, 1, 3, 4} mod 8.

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A047544 Numbers that are congruent to {1, 3, 4, 7} mod 8.

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A047600 Numbers that are congruent to {1, 3, 4, 5} mod 8.

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A369801 Maximum number of segments between n points on a circle so that they can be colored in 2 colors so that each of them intersects (at an internal point) at most one other segment of the same color.

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A109534 a(0)=1, a(n)=n+a(n-1) if n mod 2=0, a(n)=3n-a(n-1) if n mod 2 = 1.

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