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.

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

A047549 Numbers that are congruent to {0, 1, 2, 3, 4, 7} mod 8.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 8, 9, 10, 11, 12, 15, 16, 17, 18, 19, 20, 23, 24, 25, 26, 27, 28, 31, 32, 33, 34, 35, 36, 39, 40, 41, 42, 43, 44, 47, 48, 49, 50, 51, 52, 55, 56, 57, 58, 59, 60, 63, 64, 65, 66, 67, 68, 71, 72, 73, 74, 75, 76, 79, 80, 81, 82, 83, 84, 87, 88
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047420, A047602.

Programs

  • Magma
    [n : n in [0..100] | n mod 8 in [0..4] cat [7]]; // Wesley Ivan Hurt, May 29 2016
  • Maple
    A047549:=n->(24*n-33+3*cos(n*Pi)+4*sqrt(3)*cos((1-4*n)*Pi/6)+12*sin((1+
2*n)*Pi/6))/18: seq(A047549(n), n=1..100); # Wesley Ivan Hurt, Jun 16 2016
  • Mathematica
    LinearRecurrence[{1, 0, 0, 0, 0, 1, -1}, {0, 1, 2, 3, 4, 7, 8}, 50] (* G. C. Greubel, May 29 2016 *)

Formula

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*(x^5 + 3*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-33+3*cos(n*Pi)+4*sqrt(3)*cos((1-4*n)*Pi/6)+12*sin((1+
2*n)*Pi/6))/18.
a(6k) = 8k-1, 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) = (14-sqrt(2))*log(2)/16 + sqrt(2)*log(sqrt(2)+2)/8 - (2-sqrt(2))*Pi/16. - Amiram Eldar, Dec 26 2021

A047450 Numbers that are congruent to {0, 1, 2, 3, 5, 6} mod 8.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 8, 9, 10, 11, 13, 14, 16, 17, 18, 19, 21, 22, 24, 25, 26, 27, 29, 30, 32, 33, 34, 35, 37, 38, 40, 41, 42, 43, 45, 46, 48, 49, 50, 51, 53, 54, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 72, 73, 74, 75, 77, 78, 80, 81, 82, 83, 85, 86, 88
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047420, A047602.

Programs

  • Magma
    [n : n in [0..100] | n mod 8 in [0, 1, 2, 3, 5, 6]]; // Wesley Ivan Hurt, Jun 16 2016
  • Maple
    A047450:=n->(24*n-33-3*cos(n*Pi)-2*sqrt(3)*cos((1-4*n)*Pi/6)+6*sin((1+2*n) *Pi/6))/18: seq(A047450(n), n=1..100); # Wesley Ivan Hurt, Jun 16 2016
  • Mathematica
    Select[Range[0, 100], MemberQ[{0, 1, 2, 3, 5, 6}, Mod[#, 8]] &] (* Wesley Ivan Hurt, Jun 16 2016 *)

Formula

G.f.: x^2*(1+x+x^2+2*x^3+x^4+2*x^5) / ((1+x)*(1+x+x^2)*(x^2-x+1)*(x-1)^2). - R. J. Mathar, Dec 07 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-33-3*cos(n*Pi)-2*sqrt(3)*cos((1-4*n)*Pi/6)+6*sin((1+2*n) *Pi/6))/18.
a(6k) = 8k-2, 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) = 3*(sqrt(2)-1)*Pi/16 + (8-sqrt(2))*log(2)/16 + sqrt(2)*log(sqrt(2)+2)/8. - Amiram Eldar, Dec 26 2021

A250483 Numbers of the form 3^x + y^3 with x, y >= 0.

Original entry on oeis.org

1, 2, 3, 4, 9, 10, 11, 17, 27, 28, 30, 35, 36, 54, 65, 67, 73, 81, 82, 89, 91, 108, 126, 128, 134, 145, 152, 206, 217, 219, 225, 243, 244, 251, 270, 297, 307, 344, 346, 352, 368, 370, 424, 459, 513, 515, 521, 539, 586, 593, 729, 730, 732, 737, 738, 755, 756
Offset: 1

Views

Author

Vincenzo Librandi, Nov 25 2014

Keywords

Comments

No terms are congruent to 5 or 7 (mod 8): subsequence of A047420.

Examples

			17 is in this sequence because 3^2+2^3=17.
91 is in this sequence because 3^3+4^3=91.

Links

Crossrefs

Cf. A047420.
Cf. similar sequences listed in A250482.

Programs

  • Mathematica
    nn=15; Union[Select[Flatten[Table[3^x + y^3, {x, 0, nn}, {y, 0, nn}]], #<=nn^3 &]]
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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