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 1-3 of 3 results.

A047420 Numbers that are congruent to {0, 1, 2, 3, 4, 6} mod 8.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 9, 10, 11, 12, 14, 16, 17, 18, 19, 20, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 40, 41, 42, 43, 44, 46, 48, 49, 50, 51, 52, 54, 56, 57, 58, 59, 60, 62, 64, 65, 66, 67, 68, 70, 72, 73, 74, 75, 76, 78, 80, 81, 82, 83, 84, 86, 88
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

The product of any two terms belongs to the sequence and therefore also a(n)^2, a(n)^3, a(n)^4, etc. - Bruno Berselli, Nov 28 2012

Links

Crossrefs

Cf. A047549, A047602.

Programs

  • Magma
    [n : n in [0..100] | n mod 8 in [0..4] cat [6]]; // Wesley Ivan Hurt, Jun 15 2016
  • Maple
    A047420:=n->(12*n-18+sqrt(3)*(3*sin(n*Pi/3)+sin(2*n*Pi/3)))/9: seq(A047420(n), n=1..100); # Wesley Ivan Hurt, Jun 15 2016
  • Mathematica
    Select[Range[0, 100], MemberQ[{0, 1, 2, 3, 4, 6}, Mod[#, 8]] &] (* Wesley Ivan Hurt, Jun 15 2016 *)
LinearRecurrence[{2,-2,2,-2,2,-1},{0,1,2,3,4,6},70] (* Harvey P. Dale, Aug 11 2021 *)

Formula

a(n+1) = 6*floor(n/3)+(n mod 3). - Gary Detlefs, Mar 09 2010
G.f.: x^2*(1+x^2+2*x^4) / ( (1+x+x^2)*(x^2-x+1)*(x-1)^2 ). - R. J. Mathar, Dec 05 2011
From Wesley Ivan Hurt, Jun 15 2016: (Start)
a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-2*a(n-4)+2*a(n-5)-a(n-6) for n>6.
a(n) = (12*n-18+sqrt(3)*(3*sin(n*Pi/3)+sin(2*n*Pi/3)))/9.
a(6k) = 8k-2, a(6k-1) = 8k-4, a(6k-2) = 8k-5, a(6k-3) = 8k-6, a(6k-4) = 8k-7, a(6k-5) = 8k-8. (End)
Sum_{n>=2} (-1)^n/a(n) = (sqrt(2)-1)*Pi/8 + 3*log(2)/4. - Amiram Eldar, Dec 26 2021

A047602 Numbers that are congruent to {0, 1, 2, 3, 4, 5} mod 8.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 21, 24, 25, 26, 27, 28, 29, 32, 33, 34, 35, 36, 37, 40, 41, 42, 43, 44, 45, 48, 49, 50, 51, 52, 53, 56, 57, 58, 59, 60, 61, 64, 65, 66, 67, 68, 69, 72, 73, 74, 75, 76, 77, 80, 81, 82, 83, 84, 85, 88
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047420, A047549.

Programs

  • Magma
    [n: n in [0..150] | n mod 8 in [0..5]]; // Vincenzo Librandi, May 04 2016
  • Maple
    A047602:=n->floor((8/7)*floor(7*(n-1)/6)): seq(A047602(n), n=1..100); # Wesley Ivan Hurt, May 29 2016
  • Mathematica
    Table[Floor[(8/7) Floor[7 (n - 1) / 6]], {n, 80}] (* Vincenzo Librandi, May 04 2016 *)
LinearRecurrence[{1, 0, 0, 0, 0, 1, -1}, {0, 1, 2, 3, 4, 5, 8}, 50] (* G. C. Greubel, May 29 2016 *)

Formula

a(n) = floor((8/7)*floor(7*(n-1)/6)). - Bruno Berselli, May 03 2016
From Chai Wah Wu, May 29 2016: (Start)
a(n) = a(n-1) + a(n-6) - a(n-7) for n>7.
G.f.: x^2*(3*x^5 + x^4 + x^3 + x^2 + x + 1)/(x^7 - x^6 - x + 1). (End)
From Wesley Ivan Hurt, Jun 15 2016: (Start)
a(n) = (24*n-39-3*cos(n*Pi)-4*sqrt(3)*cos((4*n+1)*Pi/6)-12*sin((1-2*n)*Pi/6))/18.
a(6k) = 8k-3, a(6k-1) = 8k-4, a(6k-2) = 8k-5, a(6k-3) = 8k-6, a(6k-4) = 8k-7, a(6k-5) = 8k-8. (End)
Sum_{n>=2} (-1)^n/a(n) = sqrt(2)*Pi/16 + 7*log(2)/8 + sqrt(2)*log(3-2*sqrt(2))/16. - Amiram Eldar, Dec 26 2021

A047490 Numbers that are congruent to {0, 1, 2, 3, 5, 7} mod 8.

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 8, 9, 10, 11, 13, 15, 16, 17, 18, 19, 21, 23, 24, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 39, 40, 41, 42, 43, 45, 47, 48, 49, 50, 51, 53, 55, 56, 57, 58, 59, 61, 63, 64, 65, 66, 67, 69, 71, 72, 73, 74, 75, 77, 79, 80, 81, 82, 83, 85, 87, 88
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047450, A047549.

Programs

  • Magma
    [n : n in [0..100] | n mod 8 in [0, 1, 2, 3, 5, 7]]; // Wesley Ivan Hurt, Jun 16 2016
  • Maple
    A047490:=n->(24*n-30+6*sqrt(3)*cos((1-2*n)*Pi/6)+2*sqrt(3)*cos((1+4*n)*Pi/6))/18: seq(A047490(n), n=1..100); # Wesley Ivan Hurt, Jun 16 2016
  • Mathematica
    Select[Range[0, 100], MemberQ[{0, 1, 2, 3, 5, 7}, Mod[#, 8]] &] (* Wesley Ivan Hurt, Jun 16 2016 *)

Formula

G.f.: x^2*(x^4+x^3+x^2+1)/((x-1)^2*(x^2-x+1)*(x^2+x+1)). - Colin Barker, Jun 22 2012
From Wesley Ivan Hurt, Jun 16 2016: (Start)
a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-2*a(n-4)+2*a(n-5)-a(n-6) for n>6.
a(n) = (24*n-30+6*sqrt(3)*cos((1-2n)*Pi/6)+2*sqrt(3)*cos((1+4n)*Pi/6))/18.
a(6k) = 8k-1, a(6k-1) = 8k-3, a(6k-2) = 8k-5, a(6k-3) = 8k-6, a(6k-4) = 8k-7, a(6k-5) = 8k-8. (End)
Sum_{n>=2} (-1)^n/a(n) = (2*sqrt(2)-3)*Pi/16 + (5-sqrt(2))*log(2)/8 + sqrt(2)*log(sqrt(2)+2)/4. - Amiram Eldar, Dec 26 2021
Showing 1-3 of 3 results.

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