A047452 -id:A047452 - cp's OEIS frontend

A047409 Numbers that are congruent to {0, 1, 4, 6} mod 8.

0, 1, 4, 6, 8, 9, 12, 14, 16, 17, 20, 22, 24, 25, 28, 30, 32, 33, 36, 38, 40, 41, 44, 46, 48, 49, 52, 54, 56, 57, 60, 62, 64, 65, 68, 70, 72, 73, 76, 78, 80, 81, 84, 86, 88, 89, 92, 94, 96, 97, 100, 102, 104, 105, 108, 110, 112, 113, 116, 118, 120, 121, 124

Offset: 1

N. J. A. Sloane

All squares and the products of any terms belong to the sequence. This sequence (n > 1) is closed under multiplication. - Klaus Purath, Feb 13 2023

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

Cf. A008586, A047452.

[n : n in [0..150] | n mod 8 in [0, 1, 4, 6]]; // Wesley Ivan Hurt, May 24 2016

A047409:=n->(8*n-9-I^(2*n)+I^(-n)+I^n)/4: seq(A047409(n), n=1..100); # Wesley Ivan Hurt, May 24 2016

Select[Range[0,110], MemberQ[{0,1,4,6}, Mod[#,8]]&] (* Harvey P. Dale, Sep 28 2011 *)

G.f.: x^2*(1 + 3*x + 2*x^2 + 2*x^3) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Dec 05 2011
From Wesley Ivan Hurt, May 24 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5.
a(n) = (8*n - 9 - i^(2n) + i^(-n) + i^n)/4 where i=sqrt(-1).
a(2k) = A047452(k), a(2k-1) = A008586(k-1) for k > 0. (End)
E.g.f.: (4 + cos(x) + 4*(x - 1)*sinh(x) + (4*x - 5)*cosh(x))/2. - Ilya Gutkovskiy, May 25 2016
Sum_{n>=2} (-1)^n/a(n) = sqrt(2)*Pi/16 + (10 - sqrt(2))*log(2)/16 + sqrt(2)*log(2 + sqrt(2))/8. - Amiram Eldar, Dec 20 2021
a(n) = 2*(n-1) + floor((n+1)/4) - floor((n+2)/4). - Ridouane Oudra, Aug 19 2024

A047554 Numbers that are congruent to {1, 2, 6, 7} mod 8.

Original entry on oeis.org

1, 2, 6, 7, 9, 10, 14, 15, 17, 18, 22, 23, 25, 26, 30, 31, 33, 34, 38, 39, 41, 42, 46, 47, 49, 50, 54, 55, 57, 58, 62, 63, 65, 66, 70, 71, 73, 74, 78, 79, 81, 82, 86, 87, 89, 90, 94, 95, 97, 98, 102, 103, 105, 106, 110, 111, 113, 114, 118, 119, 121, 122, 126

Offset: 1

N. J. A. Sloane

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

Cf. A047452, A047524, A193887.

[n : n in [0..150] | n mod 8 in [1, 2, 6, 7]]; // Wesley Ivan Hurt, May 29 2016

A047554:=n->2*n+(1+I)*(2*I-2-(1-I)*I^(2*n)-I^(1-n)+I^n)/4: seq(A047554(n), n=1..100); # Wesley Ivan Hurt, May 29 2016

Select[Range[120],MemberQ[{1,2,6,7},Mod[#,8]]&] (* Harvey P. Dale, Nov 29 2011 *)

From Wesley Ivan Hurt, May 29 2016: (Start)
G.f.: x*(1+x+4*x^2+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) = 2*n+(1+i)*(2*i-2-(1-i)*i^(2*n)-i^(1-n)+i^n)/4 where i=sqrt(-1).
a(2k) = A047524(k), a(2k-1) = A047452(k). (End)
E.g.f.: (2 - sin(x) + cos(x) + (4*x - 1)*sinh(x) + (4*x - 3)*cosh(x))/2. - Ilya Gutkovskiy, May 30 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(2)*Pi/8 (A193887). - Amiram Eldar, Dec 24 2021

A047397 Numbers that are congruent to {0, 1, 2, 6} mod 8.

Original entry on oeis.org

0, 1, 2, 6, 8, 9, 10, 14, 16, 17, 18, 22, 24, 25, 26, 30, 32, 33, 34, 38, 40, 41, 42, 46, 48, 49, 50, 54, 56, 57, 58, 62, 64, 65, 66, 70, 72, 73, 74, 78, 80, 81, 82, 86, 88, 89, 90, 94, 96, 97, 98, 102, 104, 105, 106, 110, 112, 113, 114, 118, 120, 121, 122

Offset: 1

N. J. A. Sloane

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

Cf. A047452, A047467.

[n : n in [0..150] | n mod 8 in [0, 1, 2, 6]]; // Wesley Ivan Hurt, May 24 2016

A047397:=n->(8*n-11+I^(2*n)+(1+2*I)*I^(-n)+(1-2*I)*I^n)/4: seq(A047397(n), n=1..100); # Wesley Ivan Hurt, May 24 2016

Table[(8n-11+I^(2n)+(1+2*I)*I^(-n)+(1-2*I)*I^n)/4, {n, 80}] (* Wesley Ivan Hurt, May 24 2016 *)
LinearRecurrence[{1,0,0,1,-1},{0,1,2,6,8},70] (* Harvey P. Dale, Dec 31 2017 *)

G.f.: x^2*(1+x+4*x^2+2*x^3) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Dec 05 2011
From Wesley Ivan Hurt, May 24 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (8*n-11+i^(2*n)+(1+2*i)*i^(-n)+(1-2*i)*i^n)/4, where i=sqrt(-1).
a(2k) = A047452(k), a(2k-1) = A047467(k). (End)
E.g.f.: (4 + 2*sin(x) + cos(x) + (4*x - 6)*sinh(x) + (4*x - 5)*cosh(x))/2. - Ilya Gutkovskiy, May 25 2016
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 20 2021

More terms from Wesley Ivan Hurt, May 24 2016

A047439 Numbers that are congruent to {0, 1, 5, 6} mod 8.

Original entry on oeis.org

0, 1, 5, 6, 8, 9, 13, 14, 16, 17, 21, 22, 24, 25, 29, 30, 32, 33, 37, 38, 40, 41, 45, 46, 48, 49, 53, 54, 56, 57, 61, 62, 64, 65, 69, 70, 72, 73, 77, 78, 80, 81, 85, 86, 88, 89, 93, 94, 96, 97, 101, 102, 104, 105, 109, 110, 112, 113, 117, 118, 120, 121, 125

Offset: 1

N. J. A. Sloane

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

Cf. A030308, A047452, A047615.

[n : n in [0..150] | n mod 8 in [0, 1, 5, 6]]; // Wesley Ivan Hurt, May 22 2016

A047439:=n->add(gcd(i+2, i-2), i=1..n); seq(A047439(n), n=0..100); # Wesley Ivan Hurt, Jan 23 2014

Table[Sum[GCD[i + 2, i - 2], {i, n}], {n, 0, 100}] (* Wesley Ivan Hurt, Jan 23 2014 *)

a(n+1) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=1, b(1)=5 and b(k)=2^(k+1) for k>1. - Philippe Deléham, Oct 19 2011
G.f.: x^2*(1+4*x+x^2+2*x^3) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Dec 07 2011
a(n) = Sum_{i=1..n} gcd(i+2, i-2). - Wesley Ivan Hurt, Jan 23 2014
From Wesley Ivan Hurt, May 22 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = 2n+(1+i)*(4i-4-(1-i)*i^(2n)-i^(1-n)+i^n)/4 where i=sqrt(-1).
a(2n) = A047452, a(2n-1) = A047615(n). (End)
Sum_{n>=2} (-1)^n/a(n) = (2*sqrt(2)-1)*Pi/16 + (3-sqrt(2))*log(2)/8 + sqrt(2)*log(2+sqrt(2))/4. - Amiram Eldar, Dec 20 2021

More terms from Wesley Ivan Hurt, May 22 2016

A047508 Numbers that are congruent to {1, 4, 6, 7} mod 8.

Original entry on oeis.org

1, 4, 6, 7, 9, 12, 14, 15, 17, 20, 22, 23, 25, 28, 30, 31, 33, 36, 38, 39, 41, 44, 46, 47, 49, 52, 54, 55, 57, 60, 62, 63, 65, 68, 70, 71, 73, 76, 78, 79, 81, 84, 86, 87, 89, 92, 94, 95, 97, 100, 102, 103, 105, 108, 110, 111, 113, 116, 118, 119, 121, 124

Offset: 1

N. J. A. Sloane

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

Cf. A047452, A047535.

[n : n in [0..150] | n mod 8 in [1, 4, 6, 7]]; // Wesley Ivan Hurt, May 27 2016

A047508:=n->(1+I)*(4*n-4*n*I+I-1-I^(-n)+I^(1+n))/4: seq(A047508(n), n=1..100); # Wesley Ivan Hurt, May 27 2016

Table[(1+I)*(4n-4*n*I+I-1-I^(-n)+I^(1+n))/4, {n, 80}] (* Wesley Ivan Hurt, May 27 2016 *)

From Wesley Ivan Hurt, May 27 2016: (Start)
G.f.: x*(1+2*x+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)*(4*n-4*n*i+i-1-i^(-n)+i^(1+n))/4 where i=sqrt(-1).
a(2k) = A047535(k), a(2k-1) = A047452(k). (End)
E.g.f.: (2 - sin(x) - cos(x) + (4*x - 1)*exp(x))/2. - Ilya Gutkovskiy, May 27 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = (2*sqrt(2)+1)*Pi/16 + log(2)/8. - Amiram Eldar, Dec 24 2021

A047558 Numbers that are congruent to {1, 3, 6, 7} mod 8.

Original entry on oeis.org

1, 3, 6, 7, 9, 11, 14, 15, 17, 19, 22, 23, 25, 27, 30, 31, 33, 35, 38, 39, 41, 43, 46, 47, 49, 51, 54, 55, 57, 59, 62, 63, 65, 67, 70, 71, 73, 75, 78, 79, 81, 83, 86, 87, 89, 91, 94, 95, 97, 99, 102, 103, 105, 107, 110, 111, 113, 115, 118, 119, 121, 123, 126

Offset: 1

N. J. A. Sloane

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

Cf. A004767, A047452.

[n : n in [0..150] | n mod 8 in [1, 3, 6, 7]]; // Wesley Ivan Hurt, May 29 2016

A047558:=n->(8*n-3-I^(2*n)-I^(1-n)+I^(1+n))/4: seq(A047558(n), n=1..100); # Wesley Ivan Hurt, May 29 2016

Select[Range[150], MemberQ[{1,3,6,7}, Mod[#,8]]&] (* Harvey P. Dale, Jul 31 2014 *)

From Wesley Ivan Hurt, May 29 2016: (Start)
G.f.: x*(1+2*x+3*x^2+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-3-i^(2*n)-i^(1-n)+i^(1+n))/4 where i=sqrt(-1).
a(2k) = A004767(k-1) for k>0, a(2k-1) = A047452(k). (End)
E.g.f.: (2 - sin(x) + (4*x - 1)*sinh(x) + (4*x - 2)*cosh(x))/2. - Ilya Gutkovskiy, May 30 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = (2+sqrt(2))*Pi/16 + sqrt(2)*log(2+sqrt(2))/8 - (2+sqrt(2))*log(2)/16. - Amiram Eldar, Dec 24 2021

A047576 Numbers that are congruent to {1, 5, 6, 7} mod 8.

Original entry on oeis.org

1, 5, 6, 7, 9, 13, 14, 15, 17, 21, 22, 23, 25, 29, 30, 31, 33, 37, 38, 39, 41, 45, 46, 47, 49, 53, 54, 55, 57, 61, 62, 63, 65, 69, 70, 71, 73, 77, 78, 79, 81, 85, 86, 87, 89, 93, 94, 95, 97, 101, 102, 103, 105, 109, 110, 111, 113, 117, 118, 119, 121, 125

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. A047452, A047550.

[n : n in [0..150] | n mod 8 in [1, 5, 6, 7]]; // Wesley Ivan Hurt, May 29 2016

A047576:=n->(8*n-1+I^(2*n)-(2+I)*I^(-n)-(2-I)*I^n)/4: seq(A047576(n), n=1..100); # Wesley Ivan Hurt, May 29 2016

Flatten[#+{1,5,6,7}&/@(8Range[0,20])] (* Harvey P. Dale, Apr 22 2011 *)
Select[Range[100], MemberQ[{1, 5, 6, 7}, Mod[#, 8]] &] (* Vincenzo Librandi, May 30 2016 *)

From Wesley Ivan Hurt, May 29 2016: (Start)
G.f.: x*(1+4*x+x^2+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-1+i^(2*n)-(2+i)*i^(-n)-(2-i)*i^n)/4 where i=sqrt(-1).
a(2k) = A047550(k), a(2k-1) = A047452(k). (End)
E.g.f.: (2 - sin(x) - 2*cos(x) - sinh(x) + 4*x*exp(x))/2. - Ilya Gutkovskiy, May 30 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*sqrt(2)*Pi/16 - (sqrt(2)+2)*log(2)/16 + 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

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A047409 Numbers that are congruent to {0, 1, 4, 6} mod 8.

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A047554 Numbers that are congruent to {1, 2, 6, 7} mod 8.

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A047397 Numbers that are congruent to {0, 1, 2, 6} mod 8.

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A047439 Numbers that are congruent to {0, 1, 5, 6} mod 8.

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A047508 Numbers that are congruent to {1, 4, 6, 7} mod 8.

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A047558 Numbers that are congruent to {1, 3, 6, 7} mod 8.

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A047576 Numbers that are congruent to {1, 5, 6, 7} mod 8.

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