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.

Previous Showing 11-15 of 15 results.

A174015 A generalized Catalan number sequence.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 2, -3, -10, -14, -4, 34, 104, 172, 132, -197, -964, -1976, -2190, 652, 9294, 23626, 33762, 12140, -84438, -280850, -493930, -397666, 639678, 3248466, 6947462, 8068589, -2165978, -35591958, -94129444, -139864826, -56393480, 352505724
Offset: 0

Views

Author

Paul Barry, Mar 05 2010

Keywords

Comments

Hankel transform is A130151(n+1). First column of A174014.

Formula

G.f.: (sqrt(1-2x+x^2+4x^3)+3x-1)/(2x(1-x));
G.f.: 1/(1-x/(1-x/(1+x/(1-x/(1-x/(1+x/(1-... (continued fraction).
a(n) = Sum_{k, 0<=k<=n} A091866(n,k)*(-1)^(n-k) = Sum_{k, 0<=k<=n} A198379(n,k). - Philippe Deléham, Nov 27 2011
Conjecture: (n+1)*a(n) -3*n*a(n-1) +3*(n-1)*a(n-2) +3*(n-4)*a(n-3) +2*(7-2*n)*a(n-4)=0. R. J. Mathar, Nov 13 2012

Extensions

More terms from Philippe Deléham, Oct 27 2011

A263824 Permutation of the nonnegative integers: [6k+3, 6k+4, 6k+5, 6k, 6k+1, 6k+2, ...].

Original entry on oeis.org

3, 4, 5, 0, 1, 2, 9, 10, 11, 6, 7, 8, 15, 16, 17, 12, 13, 14, 21, 22, 23, 18, 19, 20, 27, 28, 29, 24, 25, 26, 33, 34, 35, 30, 31, 32, 39, 40, 41, 36, 37, 38, 45, 46, 47, 42, 43, 44, 51, 52, 53, 48, 49, 50, 57, 58, 59, 54, 55, 56, 63, 64, 65, 60, 61, 62, 69
Offset: 0

Views

Author

Wesley Ivan Hurt, Oct 27 2015

Keywords

Links

Crossrefs

Cf. A002162, A004442, A130151, A279318.

Programs

  • Magma
    [n+3*(-1)^Floor(n/3) : n in [0..100]];
  • Magma
    I:=[3,4,5,0,1]; [n le 5 select I[n] else 2*Self(n-1)- Self(n-2)-Self(n-3)+2*Self(n-4)-Self(n-5): n in [1..70]]; // Vincenzo Librandi, Nov 22 2015
  • Maple
    A263824:=n->n+3*(-1)^floor(n/3): seq(A263824(n), n=0..100);
  • Mathematica
    Table[n + 3 (-1)^Floor[n/3], {n, 0, 100}]
CoefficientList[Series[(3 - 2 x - 3 x^3 + 4 x^4)/((x - 1)^2 (1 + x^3)), {x, 0, 70}], x] (* Vincenzo Librandi, Nov 22 2015 *)
LinearRecurrence[{2,-1,-1,2,-1},{3,4,5,0,1},70] (* Harvey P. Dale, Jun 23 2017 *)
  • PARI
    Vec((3-2*x-3*x^3+4*x^4) / ((x-1)^2*(1+x^3)) + O(x^100)) \\ Altug Alkan, Oct 28 2015
  • PARI
    A263824(n)=n+3*(-1)^(n\3) \\ M. F. Hasler, Nov 25 2015

Formula

G.f.: (3-2*x-3*x^3+4*x^4) / ((x-1)^2*(1+x^3)).
a(n) = 2*a(n-1) - a(n-2) - a(n-3) + 2*a(n-4) - a(n-5), n>4.
a(n) = n + 3*(-1)^floor(n/3).
a(n) = a(n-6) + 6 for n>5. - Tom Edgar, Oct 28 2015
From Wesley Ivan Hurt, Nov 22 2015: (Start)
a(n) = n + 3*A130151(n).
a(3n) = 3*A004442(n). (End)
Sum_{n>=0, n!=3} (-1)^n/a(n) = log(2) (A002162). - Amiram Eldar, Dec 25 2023
E.g.f.: exp(-x) + x*exp(x) + 2*exp(x/2)*(cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)). - Stefano Spezia, Aug 25 2025

A132798 Period 6: repeat [0, 2, 1, 0, -2, -1].

Original entry on oeis.org

0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1
Offset: 0

Views

Author

Paul Curtz, Nov 21 2007

Keywords

Links

Crossrefs

Cf. A002264, A080425, A117373, A129339, A130151.

Programs

Formula

G.f.: x*(2+x)/((x+1)*(x^2-x+1)) = (1/3)*(4*x+1)/(x^2-x+1)-(1/3)/(x+1). - R. J. Mathar, Nov 28 2007
a(n) + a(n+1) = A117373(n+4). - R. J. Mathar, Jul 22 2009
a(n) = (-n mod 3) * (-1)^floor(n/3) = A080425(n) * (-1)^A002264(n) = A080425(n) * A130151(n). - Wesley Ivan Hurt, Jun 20 2014
From Wesley Ivan Hurt, Jun 21 2016: (Start)
a(n) + a(n-3) = 0 for n>2.
a(n) = sin(n*Pi/3) * (3*sqrt(3) + 2*sin(2*n*Pi/3))/3. (End)

A135344 a(n) = 3*a(n-1) - a(n-3) + 3*a(n-4), with initial values 1,1,1,1.

Original entry on oeis.org

1, 1, 1, 1, 5, 17, 53, 157, 469, 1405, 4217, 12653, 37961, 113881, 341641, 1024921, 3074765, 9224297, 27672893, 83018677, 249056029, 747168085, 2241504257, 6724512773, 20173538321, 60520614961, 181561844881, 544685534641, 1634056603925, 4902169811777
Offset: 0

Views

Author

Paul Curtz, Dec 06 2007

Keywords

Links

Crossrefs

Cf. A007395.

Programs

  • Mathematica
    LinearRecurrence[{3,0,-1,3},{1,1,1,1},40] (* Harvey P. Dale, Apr 15 2012 *)

Formula

3*a(n) - a(n+1) = hexaperiodic 2, 2, 2, -2, -2, -2 = 2*A130151.
From Richard Choulet, Jan 02 2008: (Start)
a(n) = (1/14)*3^n + (1/6)*(-1)^n + (16/21)*cos(Pi*n/3) + (8*sqrt(3)/21)*sin(Pi*n/3).
a(n) = (1/14)*3^n + (1/14)*[13; 11; 5; -13; -11; -5]. (End)
G.f.: ( -1+2*x+2*x^2+x^3 ) / ( (3*x-1)*(1+x)*(x^2-x+1) ). - Harvey P. Dale, Apr 15 2012
42*a(n) = 7*(-1)^n +8*A167380(n+3) +3^(n+1). - R. J. Mathar, Oct 03 2021

A275788 a(0) = 0, a(n+1) = 2*a(n) + (-1)^floor(n/3).

Original entry on oeis.org

0, 1, 3, 7, 13, 25, 49, 99, 199, 399, 797, 1593, 3185, 6371, 12743, 25487, 50973, 101945, 203889, 407779, 815559, 1631119, 3262237, 6524473, 13048945, 26097891, 52195783, 104391567, 208783133, 417566265, 835132529, 1670265059, 3340530119, 6681060239
Offset: 0

Views

Author

Paul Curtz, Aug 09 2016

Keywords

Comments

a(n) and its successive differences:
0, 1, 3, 7, 13, 25, 49, ...
1, 2, 4, 6, 12, 24, 50, 100, ...
1, 2, 2, 6, 12, 26, 50, 100, 198, ...
1, 0, 4, 6, 14, 24, 50, 98, 200, 398, ...
-1, 4, 2, 8, 10, 26, 48, 102, 198, 400, 794, ...
5, -2, 6, 2, 16, 22, 54, 96, 202, 394, 800, 1590, ...
-7, 8, -4, 14, 6, 32, 42, 106, 192, 406, 790, 1600, 3178, ...
... .
Each row has the recurrence a(n) + a(n+3) = 7*2^n.
Main diagonal: 2*A001045(n).
Upper diagonals: A084214(n+1), 3*2^n, ... .
Subdiagonals: 2^n, A078008(n), A084214(n+1), -2^n, ... .
a(-n) = 0, 1/2, 3/4, 7/8, -1/16, -17/32, -49/64, 15/128, ... .
b(n), numerators of a(-n), and first differences:
0, 1, 3, 7, -1, -17, -49, 15, 143, 399, -113, -1137, ...
1, 2, 4, -8, -16, -32, 64, 128, 256, -512, -1024, ... = A000079(n)*A130151(n), not in the OEIS.

Examples

			a(1)=2*0+1=1, a(2)=2*1+1=3, a(2)=2*3+1=7, a(3)=2*7-1=13, a(4)=2*13-1=25, ... .

Links

Crossrefs

Cf. A000079, A001045, A002264, A005009, A007283, A078008, A084214, A113405, A130151, A274817.

Programs

  • Mathematica
    CoefficientList[Series[x (1 + x + x^2)/((1 + x) (1 - 2 x) (1 - x + x^2)), {x, 0, 33}], x] (* Michael De Vlieger, Aug 11 2016 *)
LinearRecurrence[{2,0,-1,2}, {0, 1, 3, 7}, 25] (* G. C. Greubel, Aug 16 2016 *)
  • PARI
    concat(0, Vec(x*(1+x+x^2)/((1+x)*(1-2*x)*(1-x+x^2)) + O(x^40))) \\ Colin Barker, Aug 10 2016

Formula

From Colin Barker, Aug 09 2016: (Start)
a(n) = 2*a(n-1) - a(n-3) + 2*a(n-4) for n>3.
G.f.: x*(1 + x + x^2) / ((1+x)*(1-2*x)*(1-x+x^2)).
(End)
a(n+3) = 7*2^n - a(n), a(0)=0, a(1)=1, a(2)=3.

Extensions

More terms from Colin Barker, Aug 10 2016
Previous Showing 11-15 of 15 results.

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