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.

A047538 Numbers that are congruent to {0, 1, 4, 7} mod 8.

Original entry on oeis.org

0, 1, 4, 7, 8, 9, 12, 15, 16, 17, 20, 23, 24, 25, 28, 31, 32, 33, 36, 39, 40, 41, 44, 47, 48, 49, 52, 55, 56, 57, 60, 63, 64, 65, 68, 71, 72, 73, 76, 79, 80, 81, 84, 87, 88, 89, 92, 95, 96, 97, 100, 103, 104, 105, 108, 111, 112, 113, 116, 119, 120, 121, 124
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Related to a Chebyshev transform of A046055. See A074231. - Paul Barry, Oct 27 2004
Starting (1, 4, 7, ...) = partial sums of (1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, ...). - Gary W. Adamson, Jun 19 2008
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
Nonnegative m such that floor(k*(m/4)^2) = k*floor((m/4)^2), where k can assume the values from 4 to 15. See also the second comment in A047513. - Bruno Berselli, Dec 03 2015

Links

Crossrefs

Cf. A047404, A047431, A047546, A047557, A047578, A047620, A056594.
Cf. A046055, A074231.

Programs

  • Magma
    [2*n-2-(1+(-1)^n)*(-1)^((2*n-3) div 4-(-1)^n div 4) / 2 : n in [1..80]]; // Wesley Ivan Hurt, Sep 22 2015
  • Magma
    [n: n in [0..150] | n mod 8 in {0,1,4,7}]; // Vincenzo Librandi, Sep 23 2015
  • Maple
    A047538:=n->2*n-2-sin(Pi*(n-1)/2): seq(A047538(n), n=1..80); # Wesley Ivan Hurt, Sep 22 2015
  • Mathematica
    Table[2n-2-Sin[Pi*(n-1)/2], {n, 80}] (* Wesley Ivan Hurt, Sep 22 2015 *)
Select[Range[0, 150], MemberQ[{0, 1, 4, 7}, Mod[#, 8]] &] (* Vincenzo Librandi, Sep 23 2015 *)
LinearRecurrence[{2,-2,2,-1},{0,1,4,7},100] (* Harvey P. Dale, Aug 12 2016 *)
  • PARI
    a(n) = (-4+(-I)^n+I^n+4*n)/2 \\ Colin Barker, Oct 18 2015
  • PARI
    concat(0, Vec(x^2*(1+x)^2/((1+x^2)*(1-2*x+x^2)) + O(x^100))) \\ Colin Barker, Oct 18 2015
  • Sage
    [lucas_number1(n,0,1)+2*n-4 for n in (2..57)] # Zerinvary Lajos, Jul 06 2008

Formula

From Paul Barry, Oct 27 2004: (Start)
G.f.: x^2*(1+x)^2 / ((1+x^2)*(1-2*x+x^2)).
E.g.f.: 2*x*exp(x)-sin(x).
a(n) = 2*n-2-sin(Pi*(n-1)/2).
a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-a(n-4) for n>4. (End)
a(n) = 2*n-2-(1+(-1)^n)*(-1)^((2*n-3)/4-(-1)^n/4)/2. - Wesley Ivan Hurt, Sep 22 2015
a(n) = (-4+(-i)^n+i^n+4*n)/2, where i = sqrt(-1). - Colin Barker, Oct 18 2015
Sum_{n>=2} (-1)^n/a(n) = (6-sqrt(2))*log(2)/8 + sqrt(2)*log(2+sqrt(2))/4. - Amiram Eldar, Dec 20 2021

Extensions

More terms from Wesley Ivan Hurt, Sep 22 2015
G.f. adapted to offset by Colin Barker, Oct 18 2015

A047498 Numbers that are congruent to {0, 1, 2, 4, 5, 7} mod 8.

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 8, 9, 10, 12, 13, 15, 16, 17, 18, 20, 21, 23, 24, 25, 26, 28, 29, 31, 32, 33, 34, 36, 37, 39, 40, 41, 42, 44, 45, 47, 48, 49, 50, 52, 53, 55, 56, 57, 58, 60, 61, 63, 64, 65, 66, 68, 69, 71, 72, 73, 74, 76, 77, 79, 80, 81, 82, 84, 85, 87, 88
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047513, A047581.

Programs

  • Magma
    [n : n in [0..100] | n mod 8 in [0, 1, 2, 4, 5, 7]]; // Wesley Ivan Hurt, Jun 16 2016
  • Maple
    A047498:=n->(24*n-27+3*cos(n*Pi)+6*cos(n*Pi/3)+2*sqrt(3)*sin(2*n*Pi/3))/18: seq(A047498(n), n=1..100); # Wesley Ivan Hurt, Jun 16 2016
  • Mathematica
    Select[Range[0,100],MemberQ[{0,1,2,4,5,7},Mod[#,8]]&] (* or *) LinearRecurrence[{1,0,0,0,0,1,-1},{0,1,2,4,5,7,8},100] (* Harvey P. Dale, Jul 23 2015 *)

Formula

G.f.: x^2*(x^5+2*x^4+x^3+2*x^2+x+1)/((x-1)^2*(x+1)*(x^2-x+1)*(x^2+x+1)). - Colin Barker, Jun 22 2012
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-27+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-3, a(6k-2) = 8k-4, a(6k-3) = 8k-6, a(6k-4) = 8k-7, a(6k-5) = 8k-8. (End)
Sum_{n>=2} (-1)^n/a(n) = (6-3*sqrt(2))*log(2)/16 + 3*sqrt(2)*log(sqrt(2)+2)/8 - (2-sqrt(2))*Pi/16. - Amiram Eldar, Dec 27 2021

A047581 Numbers that are congruent to {0, 1, 2, 5, 6, 7} mod 8.

Original entry on oeis.org

0, 1, 2, 5, 6, 7, 8, 9, 10, 13, 14, 15, 16, 17, 18, 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33, 34, 37, 38, 39, 40, 41, 42, 45, 46, 47, 48, 49, 50, 53, 54, 55, 56, 57, 58, 61, 62, 63, 64, 65, 66, 69, 70, 71, 72, 73, 74, 77, 78, 79, 80, 81, 82, 85, 86, 87, 88
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047498, A047513.

Programs

  • Magma
    [n : n in [0..100] | n mod 8 in [0, 1, 2, 5, 6, 7]]; // Wesley Ivan Hurt, Jun 16 2016
  • Maple
    A047581:=n->(8*n+(-1)^n-2*sqrt(3)*sin(Pi*n/3)-4*sin(2*Pi*(n+1)/3)/sqrt(3)
+2*cos(Pi*n/3)-7)/6: seq(A047581(n), n=1..100); # Wesley Ivan Hurt, Jun 16 2016
  • Mathematica
    LinearRecurrence[{1, 0, 0, 0, 0, 1, -1}, {0, 1, 2, 5, 6, 7, 8}, 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^2*(x^5 + x^4 + x^3 + 3*x^2 + x + 1)/(x^7 - x^6 - x + 1). (End)
a(n) = (8*n + (-1)^n - 2*sqrt(3)*sin(Pi*n/3) - 4*sin(2*Pi*(n+1)/3)/sqrt(3) + 2*cos(Pi*n/3) - 7)/6. - Ilya Gutkovskiy, May 30 2016
a(6k) = 8k-1, a(6k-1) = 8k-2, a(6k-2) = 8k-3, a(6k-3) = 8k-6, a(6k-4) = 8k-7, a(6k-5) = 8k-8. - Wesley Ivan Hurt, Jun 16 2016
Sum_{n>=2} (-1)^n/a(n) = (12-sqrt(2))*log(2)/16 + sqrt(2)*log(sqrt(2)+2)/8 - (sqrt(2)-1)*Pi/16. - Amiram Eldar, Dec 27 2021
Showing 1-3 of 3 results.

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

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