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.

A047227 Numbers that are congruent to {1, 2, 3, 4} mod 6.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 9, 10, 13, 14, 15, 16, 19, 20, 21, 22, 25, 26, 27, 28, 31, 32, 33, 34, 37, 38, 39, 40, 43, 44, 45, 46, 49, 50, 51, 52, 55, 56, 57, 58, 61, 62, 63, 64, 67, 68, 69, 70, 73, 74, 75, 76, 79, 80, 81, 82, 85, 86, 87, 88, 91, 92, 93, 94, 97, 98
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

a(k)^m is a term for k and m in N. - Jerzy R Borysowicz, Apr 18 2023

Links

Crossrefs

Cf. A008621, A047235, A047241, A047245, A047246, A047247, A093148.
Complement of A047264. Equals A203016 divided by 3.

Programs

  • Magma
    [n: n in [0..100] | n mod 6 in [1..4]]; // Vincenzo Librandi, Jan 06 2013
  • Maple
    A047227:=n->(6*n-5-I^(2*n)+(1+I)*I^(1-n)+(1-I)*I^(n-1))/4: seq(A047227(n), n=1..100); # Wesley Ivan Hurt, May 20 2016
  • Mathematica
    Complement[Range[100], Flatten[Table[{6n - 1, 6n}, {n, 0, 15}]]] (* Alonso del Arte, Jul 07 2011 *)
Select[Range[100], MemberQ[{1, 2, 3, 4}, Mod[#, 6]]&] (* Vincenzo Librandi, Jan 06 2013 *)
  • PARI
    a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; -1,1,0,0,1]^(n-1)*[1;2;3;4;7])[1,1] \\ Charles R Greathouse IV, May 03 2023

Formula

From Johannes W. Meijer, Jul 07 2011: (Start)
a(n) = floor((n+2)/4) + floor((n+1)/4) + floor(n/4) + 2*floor((n-1)/4) + floor((n+3)/4).
G.f.: x*(1 + x + x^2 + x^3 + 2*x^4)/(x^5 - x^4 - x + 1). (End)
From Wesley Ivan Hurt, May 20 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (6n - 5 - i^(2n) + (1+i)*i^(1-n) + (1-i)*i^(n-1))/4 where i=sqrt(-1).
a(2n) = A047235(n), a(2n-1) = A047241(n). (End)
E.g.f.: (4 + sin(x) - cos(x) + (3*x - 2)*sinh(x) + 3*(x - 1)*cosh(x))/2. - Ilya Gutkovskiy, May 21 2016
From Wesley Ivan Hurt, May 21 2016: (Start)
a(n) = A047246(n) + 1.
a(n+2) - a(n+1) = A093148(n) for n>0.
a(1-n) = - A047247(n). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(3)*Pi/12 + 2*log(2)/3 - log(3)/4. - Amiram Eldar, Dec 17 2021

A047258 Numbers that are congruent to {0, 4, 5} mod 6.

Original entry on oeis.org

0, 4, 5, 6, 10, 11, 12, 16, 17, 18, 22, 23, 24, 28, 29, 30, 34, 35, 36, 40, 41, 42, 46, 47, 48, 52, 53, 54, 58, 59, 60, 64, 65, 66, 70, 71, 72, 76, 77, 78, 82, 83, 84, 88, 89, 90, 94, 95, 96, 100, 101, 102, 106, 107, 108, 112, 113, 114, 118, 119, 120, 124
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047245 (complement).

Programs

  • Magma
    [2*n-2+((2*n-2) mod 3) : n in [1..100]]; // Wesley Ivan Hurt, Apr 13 2015
  • Maple
    A047258:=n->2*n-2+((2*n-2) mod 3): seq(A047258(n), n=1..100); # Wesley Ivan Hurt, Apr 13 2015
  • Mathematica
    Flatten[#+{0,4,5}&/@(6Range[0,20])] (* Harvey P. Dale, Jul 20 2011 *)
Select[Range[0, 200], MemberQ[{0, 4, 5}, Mod[#, 6]] &] (* Vincenzo Librandi, Apr 14 2015 *)
  • PARI
    concat (0, Vec(x^2*(4+x+x^2)/((1+x+x^2)*(x-1)^2) + O(x^80))) \\ Michel Marcus, Apr 14 2015

Formula

G.f.: x^2*(4+x+x^2)/((1+x+x^2)*(x-1)^2). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, Apr 13 2015: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.
a(n) = 2n-2 + ((2n-2) mod 3). (End)
From Wesley Ivan Hurt, Jun 13 2016: (Start)
a(n) = 2*n-1-2*sin(2*n*Pi/3)/sqrt(3).
a(3k) = 6k-1, a(3k-1) = 6k-2, a(3k-2) = 6k-6. (End)
E.g.f.: 1 + (2*x - 1)*exp(x) - 2*sin(sqrt(3)*x/2)*(cosh(x/2) - sinh(x/2))/sqrt(3). - Ilya Gutkovskiy, Jun 14 2016
Sum_{n>=2} (-1)^n/a(n) = log(2+sqrt(3))/(2*sqrt(3)) - (3-sqrt(3))*Pi/18. - Amiram Eldar, Dec 14 2021

Extensions

More terms from Wesley Ivan Hurt, Apr 13 2015

A267068 a(n) = (n+1) / A189733(n).

Original entry on oeis.org

1, 2, 3, 2, 5, 1, 7, 2, 3, 2, 11, 1, 13, 2, 3, 2, 17, 1, 19, 2, 3, 2, 23, 1, 25, 2, 3, 2, 29, 1, 31, 2, 3, 2, 35, 1, 37, 2, 3, 2, 41, 1, 43, 2, 3, 2, 47, 1, 49, 2, 3, 2, 53, 1, 55, 2, 3, 2, 59, 1, 61, 2, 3, 2, 65, 1, 67, 2, 3, 2
Offset: 0

Views

Author

Paul Curtz, Jan 10 2016

Keywords

Comments

A189733(n) is the denominator of an autosequence of the first kind (the main diagonal is A000004).

Crossrefs

Cf. A000004, A047238, A047245, A052901, A130196, A189733.

Programs

  • Mathematica
    CoefficientList[Series[(1 + 2 x + 3 x^2 + 2 x^3 + 5 x^4 + x^5 + 5 x^6 - 2 x^7 - 3 x^8 - 2 x^9 + x^10 - x^11)/((1 - x)^2 (1 + x)^2 (1 - x + x^2)^2 (1 + x + x^2)^2), {x, 0, 69}], x] (* or *)
b[m_, n_] := b[m, n] = Which[m == n, 0, n == m + 1, (-1)^(n + 1)/n, n > m, b[m, n - 1] + b[m + 1, n - 1], n < m, b[m - 1, n + 1] - b[m - 1, n]]; Table[(n + 1)/Denominator@ b[0, n], {n, 0, 69}] (* Michael De Vlieger, Jan 15 2016, Jean-François Alcover at A189733 *)

Formula

a(2n+1) = A130196(n+1).
A052901(n+2) = period 3: 2, 3, 2 is at rank A047245(n+1) = 1, 2, 3, 7, 8, 9, ... .
Conjectures from Colin Barker, Jan 10 2016: (Start)
a(n) = 2*a(n-6) - a(n-12) for n>11.
G.f.: (1+2*x+3*x^2+2*x^3+5*x^4+x^5+5*x^6-2*x^7-3*x^8-2*x^9+x^10-x^11) / ((1-x)^2*(1+x)^2*(1-x+x^2)^2*(1+x+x^2)^2).
(End)
a(3n) + a(3n+1) + a(3n+2) = A047238(n+3).
Showing 1-3 of 3 results.

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

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