cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A047451 Numbers that are congruent to {0, 6} mod 8.

Original entry on oeis.org

0, 6, 8, 14, 16, 22, 24, 30, 32, 38, 40, 46, 48, 54, 56, 62, 64, 70, 72, 78, 80, 86, 88, 94, 96, 102, 104, 110, 112, 118, 120, 126, 128, 134, 136, 142, 144, 150, 152, 158, 160, 166, 168, 174, 176, 182, 184, 190, 192, 198, 200, 206, 208, 214, 216, 222, 224, 230
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

All even numbers m such that Integral_{x=0..2*Pi} Product_{i=1..m/2} cos(2*i*x) dx is nonzero. - William Boyles, Oct 12 2019

Links

Crossrefs

Union of A008590 and A017137.
Cf. A030308, A047504 (complement).

Programs

  • Mathematica
    Array[8 # + {0, 6} &, 29, 0] // Flatten (* or *)
Rest@ CoefficientList[Series[2 x^2*(3 + x)/((1 + x) (x - 1)^2), {x, 0, 58}], x] (* Michael De Vlieger, Nov 18 2019 *)
LinearRecurrence[{1,1,-1},{0,6,8},80] (* Harvey P. Dale, Apr 09 2022 *)
  • PARI
    forstep(n=0,200,[6,2],print1(n", ")) \\ Charles R Greathouse IV, Oct 17 2011
  • PARI
    a(n) = 4*n - 3 + (-1)^n; \\ David Lovler, Jul 25 2022

Formula

a(n) = 8*n - a(n-1) - 10 (with a(1)=0). - Vincenzo Librandi, Aug 06 2010
From R. J. Mathar, Oct 08 2011: (Start)
a(n) = 4*n - 3 + (-1)^n.
G.f.: 2*x^2*(3+x) / ( (1+x)*(x-1)^2 ). (End)
a(n+1) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=6 and b(k)=2^(k+2) for k > 0. - Philippe Deléham, Oct 17 2011
a(n) = ceiling((8/3)*ceiling(3*n/2)). - Clark Kimberling, Jul 04 2012
Sum_{n>=2} (-1)^n/a(n) = 3*log(2)/8 - Pi/16. - Amiram Eldar, Dec 18 2021
E.g.f.: (4*x + 1)*exp(x) - exp(-x) = 4*x*exp(x) + 2*sinh(x). - David Lovler, Aug 02 2022

A047422 Numbers that are congruent to {1, 2, 3, 4, 5, 6} mod 8.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 25, 26, 27, 28, 29, 30, 33, 34, 35, 36, 37, 38, 41, 42, 43, 44, 45, 46, 49, 50, 51, 52, 53, 54, 57, 58, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70, 73, 74, 75, 76, 77, 78, 81, 82, 83, 84, 85, 86
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047504, A047519.

Programs

  • Magma
    [n : n in [0..100] | n mod 8 in [1..6]]; // Wesley Ivan Hurt, Jun 16 2016
  • Maple
    A047422:=n->(24*n-21-3*cos(n*Pi)-4*sqrt(3)*cos((1+4*n)*Pi/6)-12*sin((1-2*n)*Pi/6))/18: seq(A047422(n), n=1..100); # Wesley Ivan Hurt, Jun 16 2016
  • Mathematica
    Select[Range[0, 100], MemberQ[{1, 2, 3, 4, 5, 6}, Mod[#, 8]] &] (* Wesley Ivan Hurt, Jun 16 2016 *)

Formula

G.f.: x*(1+x+x^2+x^3+x^4+x^5+2*x^6) / ((1+x)*(1+x+x^2)*(x^2-x+1)*(x-1)^2). - R. J. Mathar, Dec 05 2011
From Wesley Ivan Hurt, Jun 16 2016: (Start)
a(n) = a(n-1) + a(n-6) - a(n-7) for n>7.
a(n) = (24*n-21-3*cos(n*Pi)-4*sqrt(3)*cos((1+4*n)*Pi/6)-12*sin((1-2*n)*Pi/6))/18.
a(6k) = 8k-2, a(6k-1) = 8k-3, a(6k-2) = 8k-4, a(6k-3) = 8k-5, a(6k-4) = 8k-6, a(6k-5) = 8k-7. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)+1)*Pi/16 + log(2)/2 + sqrt(2)*log(3-2*sqrt(2))/16. - Amiram Eldar, Dec 28 2021

A047519 Numbers that are congruent to {1, 2, 3, 4, 6, 7} mod 8.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 10, 11, 12, 14, 15, 17, 18, 19, 20, 22, 23, 25, 26, 27, 28, 30, 31, 33, 34, 35, 36, 38, 39, 41, 42, 43, 44, 46, 47, 49, 50, 51, 52, 54, 55, 57, 58, 59, 60, 62, 63, 65, 66, 67, 68, 70, 71, 73, 74, 75, 76, 78, 79, 81, 82, 83, 84, 86, 87
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047422, A047504.

Programs

  • Magma
    [n : n in [0..100] | n mod 8 in [1, 2, 3, 4, 6, 7]]; // Wesley Ivan Hurt, Jun 16 2016
  • Maple
    A047519:=n->(24*n-15-3*cos(n*Pi)-2*sqrt(3)*cos((1-4*n)*Pi/6)+6*sin((1+2*n)
*Pi/6))/18: seq(A047519(n), n=1..100); # Wesley Ivan Hurt, Jun 16 2016
  • Mathematica
    LinearRecurrence[{1, 0, 0, 0, 0, 1, -1}, {1, 2, 3, 4, 6, 7, 9}, 50] (* G. C. Greubel, May 30 2016 *)

Formula

From Chai Wah Wu, May 30 2016: (Start)
a(n) = a(n-1) + a(n-6) - a(n-7), for n > 7.
G.f.: x*(x^6 + x^5 + 2*x^4 + x^3 + x^2 + x + 1)/(x^7 - x^6 - x + 1). (End)
From Wesley Ivan Hurt, Jun 16 2016: (Start)
a(n) = (24*n-15-3*cos(n*Pi)-2*sqrt(3)*cos((1-4*n)*Pi/6)+6*sin((1+2*n)*Pi/6))/18.
a(6k) = 8k-1, a(6k-1) = 8k-2, a(6k-2) = 8k-4, a(6k-3) = 8k-5, a(6k-4) = 8k-6, a(6k-5) = 8k-7. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (3*sqrt(2)-1)*Pi/16 + log(2)/4 + sqrt(2)*log(3-2*sqrt(2))/16. - Amiram Eldar, Dec 28 2021

A047572 Numbers that are congruent to {1, 2, 4, 5, 6, 7} mod 8.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 9, 10, 12, 13, 14, 15, 17, 18, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 36, 37, 38, 39, 41, 42, 44, 45, 46, 47, 49, 50, 52, 53, 54, 55, 57, 58, 60, 61, 62, 63, 65, 66, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 81, 82, 84, 85, 86, 87
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047422, A047504, A047519.

Programs

Formula

From Wesley Ivan Hurt, Jun 16 2016: (Start)
G.f.: x*(1+x+2*x^2+x^3+x^4+x^5+x^6) / ((x-1)^2*(1+x+x^2+x^3+x^4+x^5)).
a(n) = a(n-1) + a(n-6) - a(n-7) for n>7.
a(n) = (24*n-9-3*cos(n*Pi)-6*cos(n*Pi/3)+2*sqrt(3)*sin(2*n*Pi/3))/18.
a(6k) = 8k-1, a(6k-1) = 8k-2, a(6k-2) = 8k-3, a(6k-3) = 8k-4, a(6k-4) = 8k-6, a(6k-5) = 8k-7. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (3*sqrt(2)-1)*Pi/16 + sqrt(2)*log(sqrt(2)+2)/8 - (sqrt(2)+4)*log(2)/16. - Amiram Eldar, Dec 28 2021
Showing 1-4 of 4 results.

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