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.

A168582 a(n) = (4*n^3 - 6*n^2 + 8*n + 9 + 3*(-1)^n)/12.

Original entry on oeis.org

1, 1, 3, 7, 17, 33, 59, 95, 145, 209, 291, 391, 513, 657, 827, 1023, 1249, 1505, 1795, 2119, 2481, 2881, 3323, 3807, 4337, 4913, 5539, 6215, 6945, 7729, 8571, 9471, 10433, 11457, 12547, 13703, 14929, 16225, 17595, 19039, 20561
Offset: 0

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Author

Paul Curtz, Nov 30 2009

Keywords

Comments

Starting with a(2), the sum of the first and last term in row n-1 of the Janet table A172002.

Links

Crossrefs

Cf. A137928 (first differences).

Programs

  • Magma
    [2*n/3 +3/4 -n^2/2 +n^3/3 +(-1)^n/4: n in [0..40]]; // Vincenzo Librandi, Aug 06 2011
  • Mathematica
    Table[(4*n^3 - 6*n^2 + 8*n + 9 + 3*(-1)^n)/12, {n,0,50}] (* G. C. Greubel, Jul 26 2016 *)
  • PARI
    a(n)=(4*n^3-6*n^2+8*n+9+3*(-1)^n)/12 \\ Charles R Greathouse IV, Jul 26 2016

Formula

a(n+2) = A168388(n) + A168380(n), n >= 0.
a(2n) = A168547(n);
a(2n+1) = A168574(n).
G.f.: (1 - 2*x + x^4 + 2*x^2 + 2*x^3)/((1+x)*(x-1)^4). - R. J. Mathar, Jun 27 2011
E.g.f.: (1/12)*((4*x^3 + 6*x^2 + 6*x + 9)*exp(x) + 3*exp(-x)). - G. C. Greubel, Jul 26 2016

A171219 A138101(n)+A168142(n).

Original entry on oeis.org

5, 5, 21, 21, 21, 21, 21, 21, 21, 21, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121
Offset: 1

Views

Author

Paul Curtz, Dec 05 2009

Keywords

Comments

This here basically repeats entries A168388(2k+1) 2*k^2 times for k=1,2,....
The construction is similar to A168234.

Extensions

Comments tightened - R. J. Mathar, Nov 24 2010

A272000 Coinage sequence: a(n) = A018227(n)-7.

Original entry on oeis.org

3, 11, 29, 47, 79, 111, 161, 211, 283, 355, 453, 551, 679, 807, 969, 1131, 1331, 1531, 1773, 2015, 2303, 2591, 2929, 3267, 3659, 4051, 4501, 4951, 5463, 5975, 6553, 7131, 7779, 8427, 9149, 9871, 10671, 11471, 12353, 13235, 14203, 15171, 16229, 17287, 18439
Offset: 1

Views

Author

Natan Arie Consigli, Jul 02 2016

Keywords

Comments

Terms from 29 to 111 are the atomic numbers of the elements of group 11 in the periodic table. The group is also known as the coinage metals since copper (element 29), silver (element 47) and gold (element 79) are in group 11.

Links

Crossrefs

Other groups: 1(A219527), 2(A168380), 3(A168388), 12(A271998), 13(A271997), 14(A271996), 15(A271995), 16(A271994), 17(A271999), 18(A018227).

Programs

  • Mathematica
    LinearRecurrence[{2,1,-4,1,2,-1},{3,11,29,47,79,111},50] (* Harvey P. Dale, Nov 26 2018 *)
  • PARI
    Vec(x*(3+5*x+4*x^2-10*x^3-3*x^4+5*x^5)/((1-x)^4*(1+x)^2) + O(x^60)) \\ Colin Barker, Oct 25 2016

Formula

From Colin Barker, Oct 25 2016: (Start)
G.f.: x*(3 + 5*x + 4*x^2 - 10*x^3 - 3*x^4 + 5*x^5)/((1 - x)^4*(1 + x)^2).
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n>6.
a(n) = (n^3 + 9*n^2 + 26*n - 30)/6 for n even.
a(n) = (n^3 + 9*n^2 + 29*n - 21)/6 for n odd. (End)
Showing 1-3 of 3 results.

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