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.

A299120 a(n) = (n-1)*(n-2)*(n+3)*(n+2)/12.

Original entry on oeis.org

1, 0, 0, 5, 21, 56, 120, 225, 385, 616, 936, 1365, 1925, 2640, 3536, 4641, 5985, 7600, 9520, 11781, 14421, 17480, 21000, 25025, 29601, 34776, 40600, 47125, 54405, 62496, 71456, 81345, 92225, 104160, 117216, 131461, 146965, 163800, 182040, 201761, 223041
Offset: 0

Views

Author

Juri-Stepan Gerasimov, Feb 03 2018

Keywords

Links

Crossrefs

Cf. A001263, A033275, A077415, A299146, A299198.

Programs

  • GAP
    List([0..10^3], n->n^4/12+n^3/6-7*n^2/12-2*n/3+1); # Muniru A Asiru, Feb 04 2018
  • Magma
    [n^4/12 + n^3/6 - 7*n^2/12 - 2*n/3 + 1: n in [0..40]];
  • Maple
    seq(n^4/12+n^3/6-7*n^2/12-2*n/3+1, n=0..10^3); # Muniru A Asiru, Feb 04 2018
  • Mathematica
    Rest@ CoefficientList[Series[(1 - 5 x + 10 x^2 - 5 x^3 + x^4)/(1 - x)^5, {x, 0, 41}], x] (* Michael De Vlieger, Feb 10 2018 *)
f[n_] := n^4/12 + n^3/6 - 7*n^2/12 - 2*n/3 + 1; Array[f, 40, 0] (* or *)
LinearRecurrence[{5, -10, 10, -5, 1}, {1, 0, 0, 5, 21}, 40] (* Robert G. Wilson v, Mar 12 2018 *)
  • PARI
    Vec((1 - 5*x + 10*x^2 - 5*x^3 + x^4) / (1 - x)^5 + O(x^50)) \\ Colin Barker, Feb 05 2018

Formula

a(n) = n^4/12 + n^3/6 - 7*n^2/12 - 2*n/3 + 1 = (n-1)*(n-2)*(n+3)*(n+2)/12.
From Colin Barker, Feb 05 2018: (Start)
G.f.: (1 - 5*x + 10*x^2 - 5*x^3 + x^4) / (1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5. (End)
a(n) = A033275(n+2) for n > 1. - Georg Fischer, Oct 09 2018
From Amiram Eldar, Jan 12 2021: (Start)
Sum_{n>=3} 1/a(n) = 43/150.
Sum_{n>=3} (-1)^(n+1)/a(n) = 16*log(2)/5 - 154/75. (End)
E.g.f.: exp(x)*(12 - 12*x + 6*x^2 + 8*x^3 + x^4)/12. - Stefano Spezia, Feb 21 2024

Extensions

Edited by Wolfdieter Lang, Apr 06 2018

A299198 a(n) = n^4/6 - 2*n^3/3 - n^2/6 + 5*n/3 + 1.

Original entry on oeis.org

2, 1, 0, 5, 26, 77, 176, 345, 610, 1001, 1552, 2301, 3290, 4565, 6176, 8177, 10626, 13585, 17120, 21301, 26202, 31901, 38480, 46025, 54626, 64377, 75376, 87725, 101530, 116901, 133952, 152801, 173570, 196385, 221376, 248677, 278426, 310765, 345840, 383801, 424802, 469001
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Feb 04 2018

Keywords

Examples

			For n=2, a(2) = 1^4/6 - 2*1^3/3 - 1^2/6 + 5*1/3 + 1 = 2.

Links

Crossrefs

Cf. A001263, A067998, A077415, A299120, A299146.

Programs

  • GAP
    List([1..50], n -> n^4/6-2*n^3/3-n^2/6+5*n/3+1); # Muniru A Asiru, Feb 04 2018
  • Julia
    [div((n-3)*(n+1)*(n^2-2*n-2),6) for n in 1:50] |> println # Bruno Berselli, Apr 11 2018
  • Magma
    [n^4/6-2*n^3/3-n^2/6+5*n/3+1: n in [1..50]];
  • Maple
    seq(n^4/6-2*n^3/3-n^2/6+5*n/3+1,n=1..50); # Muniru A Asiru, Feb 04 2018
  • Mathematica
    f[n_] := n^4/6 - 2 n^3/3 - n^2/6 + 5 n/3 + 1; Array[f, 50] (* or *)
CoefficientList[ Series[(-2 + 9 x - 15 x^2 + 5 x^3 - x^4)/(-1 + x)^5, {x, 0, 50}], x] (* or *)
LinearRecurrence[{5, -10, 10, -5, 1}, {2, 1, 0, 5, 26}, 50] (* Robert G. Wilson v, Feb 09 2018 *)
  • PARI
    Vec(x*(2 - 9*x + 15*x^2 - 5*x^3 + x^4) / (1 - x)^5 + O(x^50)) \\ Colin Barker, Feb 05 2018

Formula

a(n) = (n - 3)*(n + 1)*(n^2 - 2*n - 2)/6 = A299120(n-1) + A299120(1-n).
From Colin Barker, Feb 05 2018: (Start)
G.f.: x*(2 - 9*x + 15*x^2 - 5*x^3 + x^4) / (1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5.
(End)
E.g.f.: exp(x)*(6 + 6*x - 6*x^2 + 2*x^3 + x^4)/6. - Iain Fox, Feb 09 2018
6*a(n) = A067998(n)^2 - 5*A067998(n) + 6. - Bruno Berselli, Apr 11 2018
Showing 1-2 of 2 results.

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