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.

A047238 Numbers that are congruent to {0, 2} mod 6.

Original entry on oeis.org

0, 2, 6, 8, 12, 14, 18, 20, 24, 26, 30, 32, 36, 38, 42, 44, 48, 50, 54, 56, 60, 62, 66, 68, 72, 74, 78, 80, 84, 86, 90, 92, 96, 98, 102, 104, 108, 110, 114, 116, 120, 122, 126, 128, 132, 134, 138, 140, 144, 146, 150, 152, 156, 158, 162
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

Complement of A047251, or "Polyrhythmic Sequence" P(2,3); the present sequence represents where the "rests" occur in a "3 against 2" polyrhythm. (See A267027 for definition and description). - Bob Selcoe, Jan 12 2016

Links

Crossrefs

Cf. A047270 [(6*n-(-1)^n-1)/2], A047235 [(6*n-(-1)^n-3)/2], A047241 [(6*n-(-1)^n-5)/2].
Cf. A047251, A267027.

Programs

  • Magma
    [n: n in [0..200]|n mod 6 in {0,2}]; // Vincenzo Librandi, Jan 12 2016
  • Mathematica
    Select[Range[0,200],MemberQ[{0,2},Mod[#,6]]&] (* or *) LinearRecurrence[ {1,1,-1},{0,2,6},70] (* Harvey P. Dale, Jun 15 2011 *)
  • PARI
    forstep(n=0,200,[2,4],print1(n", ")) \\ Charles R Greathouse IV, Oct 17 2011

Formula

From Bruno Berselli, Jun 24 2010: (Start)
G.f.: 2*x*(1+2*x)/((1+x)*(1-x)^2).
a(n) = a(n-1) + a(n-2) - a(n-3), a(0)=0, a(1)=2, a(2)=6.
a(n) = (6*n - (-1)^n-7)/2.
a(n) = 2*A032766(n-1). (End)
a(n) = 6*n - a(n-1) - 10 (with a(1)=0). - Vincenzo Librandi, Aug 05 2010
a(n+1) = Sum_{k>=0} A030308(n,k)*A111286(k+2). - Philippe Deléham, Oct 17 2011
a(n) = 2*floor(3*n/2). - Enrique Pérez Herrero, Jul 04 2012
Sum_{n>=2} (-1)^n/a(n) = sqrt(3)*Pi/36 + log(3)/4. - Amiram Eldar, Dec 13 2021
E.g.f: 3*(x-1)*exp(x) - cosh(x) + 4. - David Lovler, Jul 11 2022

A047255 Numbers that are congruent to {1, 2, 3, 5} mod 6.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 9, 11, 13, 14, 15, 17, 19, 20, 21, 23, 25, 26, 27, 29, 31, 32, 33, 35, 37, 38, 39, 41, 43, 44, 45, 47, 49, 50, 51, 53, 55, 56, 57, 59, 61, 62, 63, 65, 67, 68, 69, 71, 73, 74, 75, 77, 79, 80, 81, 83, 85, 86, 87, 89, 91, 92, 93, 95, 97, 98, 99, 101, 103, 104
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Each element is coprime to preceding two elements. - Amarnath Murthy, Jun 12 2001
The sequence is the interleaving of A047241 with A016789. - Guenther Schrack, Feb 16 2019

Examples

			After 21 and 23 the next term is 25 as 24 has a common divisor with 21.

Links

Crossrefs

Cf. A007310, A016789, A047228, A047237, A047241, A047251, A047261, A047273, A062062, A062063.
Complement: A047233

Programs

  • Haskell
    a047255 n = a047255_list !! (n-1)
a047255_list = 1 : 2 : 3 : 5 : map (+ 6) a047255_list
-- Reinhard Zumkeller, Jan 17 2014
  • Magma
    [n : n in [0..100] | n mod 6 in [1, 2, 3, 5]]; // Wesley Ivan Hurt, May 21 2016
  • Maple
    A047255:=n->(6*n-4+I^(1-n)+I^(n-1))/4: seq(A047255(n), n=1..100); # Wesley Ivan Hurt, May 20 2016
  • Mathematica
    Select[Range[100], MemberQ[{1, 2, 3, 5}, Mod[#, 6]] &]
LinearRecurrence[{2,-2,2,-1},{1,2,3,5},100] (* Harvey P. Dale, May 14 2020 *)
  • PARI
    a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,2,-2,2]^(n-1)*[1;2;3;5])[1,1] \\ Charles R Greathouse IV, Feb 11 2017
  • Sage
    a=(x*(1+x^2+x^3)/((1+x^2)*(1-x)^2)).series(x, 80).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Feb 16 2019

Formula

{k | k == 1, 2, 3, 5 (mod 6)}.
G.f.: x*(1 + x^2 + x^3) / ((1+x^2)*(1-x)^2). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, May 20 2016: (Start)
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4), for n>4.
a(n) = (6*n - 4 + i^(1-n) + i^(n-1))/4, where i = sqrt(-1).
a(2*n) = A016789(n-1) for n>0, a(2*n-1) = A047241(n). (End)
E.g.f.: (2 + sin(x) + (3*x - 2)*exp(x))/2. - Ilya Gutkovskiy, May 21 2016
a(1-n) = - A047251(n). - Wesley Ivan Hurt, May 21 2016
From Guenther Schrack, Feb 16 2019: (Start)
a(n) = (6*n - 4 + (1 - (-1)^n)*(-1)^(n*(n-1)/2))/4.
a(n) = a(n-4) + 6, a(1)=1, a(2)=2, a(3)=3, a(4)=5, for n > 4.
a(n) = A047237(n) + 1. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = 5*sqrt(3)*Pi/36 + log(2)/3 - log(3)/4. - Amiram Eldar, Dec 17 2021
a(n) = 2*n - 1 - floor(n/2) + floor(n/4) - floor((n+1)/4). - Ridouane Oudra, Feb 21 2023

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Jun 15 2001

A102214 Expansion of (1 + 4*x + 4*x^2)/((1+x)*(1-x)^3).

Original entry on oeis.org

1, 6, 16, 30, 49, 72, 100, 132, 169, 210, 256, 306, 361, 420, 484, 552, 625, 702, 784, 870, 961, 1056, 1156, 1260, 1369, 1482, 1600, 1722, 1849, 1980, 2116, 2256, 2401, 2550, 2704, 2862, 3025, 3192, 3364, 3540, 3721, 3906, 4096, 4290, 4489, 4692, 4900
Offset: 0

Views

Author

Creighton Dement, Feb 17 2005

Keywords

Comments

A floretion-generated sequence.
a(n) gives the number of triples (x,y,x+y) with positive integers satisfying x < y and x + y <= 3*n. - Marcus Schmidt (marcus-schmidt(AT)gmx.net), Jan 13 2006
Number of different partitions of numbers x + y = z such that {x,y,z} are integers {1,2,3,...,3n} and z > y > x. - Artur Jasinski, Feb 09 2010
Second bisection preceded by zero is A152743. - Bruno Berselli, Oct 25 2011
a(n) has no final digit 3, 7, 8. - Paul Curtz, Mar 04 2020
One odd followed by three evens.
From Paul Curtz, Mar 06 2020: (Start)
b(n) = 0, 1, 6, 16, 30, 49, ... = 0, a(n).
( 25, 12, 4, 0, 1, 6, 16, 30, ...
-13, -8, -4 1, 5, 10, 14, 19, ...
5, 4, 5, 4, 5, 4, 5, 4, ... .)
b(-n) = 0, 4, 12, 25, 42, 64, 90, 121, ... .
A154589(n) are in the main diagonal of b(n) and b(-n). (End)

Links

Crossrefs

Cf. A016778, A038764, A001859, A006578, A069905, A077043, A274221, A330707.
Cf. A000326, A016789, A152743, A001651, A032766, A047237, A047251, A154589.

Programs

  • Magma
    [(6*n*(3*n+4)+(-1)^n+7)/8: n in [0..60]]; // Vincenzo Librandi, Oct 26 2011
  • Mathematica
    aa = {}; Do[i = 0; Do[Do[Do[If[x + y == z, i = i + 1], {x, y + 1, 3 n}], {y, 1, 3 n}], {z, 1, 3 n}]; AppendTo[aa, i], {n, 1, 20}]; aa (* Artur Jasinski, Feb 09 2010 *)
  • PARI
    a(n)=(6*n*(3*n+4)+(-1)^n+7)/8 \\ Charles R Greathouse IV, Apr 16 2020

Formula

G.f.: -(4*x^2 + 4*x + 1)/((x+1)*(x-1)^3) = (1+2*x)^2/((1+x)*(1-x)^3).
a(2n) = A016778(n) = (3n+1)^2.
a(n) + a(n+1) = A038764(n+1).
a(n) = floor( (3*n+2)/2 ) * ceiling( (3*n+2)/2 ). - Marcus Schmidt (marcus-schmidt(AT)gmx.net), Jan 13 2006
a(n) = (6*n*(3*n+4) + (-1)^n+7)/8. - Bruno Berselli, Oct 25 2011
a(n) = A198392(n) + A198392(n-1). - Bruno Berselli, Nov 06 2011
From Paul Curtz, Mar 04 2020: (Start)
a(n) = A006578(n) + A001859(n) + A077043(n+1).
a(n) = A274221(2+2*n).
a(20+n) - a(n) = 30*(32+3*n).
a(1+2*n) = 3*(1+n)*(2+3*n).
a(n) = A047237(n) * A047251(n).
a(n) = A001651(n+1) * A032766(n).(End)
E.g.f.: ((4 + 21*x + 9*x^2)*cosh(x) + 3*(1 + 7*x + 3*x^2)*sinh(x))/4. - Stefano Spezia, Mar 04 2020

Extensions

Definition rewritten by Bruno Berselli, Oct 25 2011
Showing 1-3 of 3 results.

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

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