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 1-2 of 2 results.

A099530 Expansion of 1/(1 - x + x^4).

Original entry on oeis.org

1, 1, 1, 1, 0, -1, -2, -3, -3, -2, 0, 3, 6, 8, 8, 5, -1, -9, -17, -22, -21, -12, 5, 27, 48, 60, 55, 28, -20, -80, -135, -163, -143, -63, 72, 235, 378, 441, 369, 134, -244, -685, -1054, -1188, -944, -259, 795, 1983, 2927, 3186, 2391, 408, -2519, -5705, -8096, -8504, -5985, -280, 7816, 16320, 22305, 22585, 14769
Offset: 0

Views

Author

Paul Barry, Oct 20 2004

Keywords

Comments

Binomial transform of A099531.

Links

Crossrefs

Cf. A050935, A099531.

Programs

  • Magma
    [n le 4 select 1 else Self(n-1) -Self(n-4): n in [1..81]]; // G. C. Greubel, Apr 13 2023
  • Mathematica
    LinearRecurrence[{1,0,0,-1}, {1,1,1,1}, 80] (* G. C. Greubel, Apr 13 2023 *)
  • SageMath
    @CachedFunction
def a(n): # a = A099530
    if (n<4): return 1
    else: return a(n-1) - a(n-4)
[a(n) for n in range(81)] # G. C. Greubel, Apr 13 2023

Formula

a(n) = a(n-1) - a(n-4).
a(n) = Sum_{k=0..floor(n/4)} binomial(n-3*k, k)*(-1)^k.

A193885 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) - a(n-4), n>=4; a(0) = 1, a(1) = 1, a(2) = 2, a(3) = 3.

Original entry on oeis.org

1, 1, 2, 3, 3, 1, -5, -18, -41, -75, -115, -143, -118, 35, 431, 1213, 2499, 4254, 6047, 6665, 3609, -7375, -32334, -77933, -147781, -234503, -305765, -283634, -20329, 718653, 2239077, 4824577, 8495482, 12533139, 14698471, 10166901, -9557053, -57006530
Offset: 0

Views

Author

Johannes W. Meijer, Aug 11 2011

Keywords

Comments

The Ze1 sums, see A180662, of triangle A108299 equal the terms of this sequence.

Links

Crossrefs

Cf. A099531, A108299, A180662.

Programs

  • Magma
    I:=[1, 1, 2, 3 ]; [n le 4 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jul 10 2012
  • Maple
    A193885 := proc(n) option remember: if n=0 then 1 elif n=1 then 1 elif n=2 then 2 elif n=3 then 3 elif n>=4 then 3*procname(n-1)-3*procname(n-2)+procname(n-3)-procname(n-4) fi: end: seq(A193885(n),n=0..37);
  • Mathematica
    CoefficientList[Series[(1-x)*(1-x+x^2)/(1-3*x+3*x^2-x^3+x^4),{x,0,50}],x] (* Vincenzo Librandi, Jul 10 2012 *)

Formula

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) - a(n-4), n>=4; a(0) = 1, a(1) = 1, a(2) = 2, a(3) = 3.
G.f.: (1-x)*(1-x+x^2)/(1-3*x+3*x^2-x^3+x^4).
a(n) = (-1)^(n+1)*(A099531(n+4) + 2*A099531(n+3) + 2*A099531(n+2) + A099531(n+1)).
Showing 1-2 of 2 results.

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