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.

A242727 Sum of the third largest parts of the partitions of 4n into 4 parts.

Original entry on oeis.org

1, 7, 29, 86, 198, 396, 719, 1203, 1899, 2866, 4156, 5840, 7997, 10695, 14025, 18086, 22962, 28764, 35611, 43603, 52871, 63554, 75768, 89664, 105401, 123111, 142965, 165142, 189790, 217100, 247271, 280467, 316899, 356786, 400308, 447696, 499189, 554983
Offset: 1

Views

Author

Wesley Ivan Hurt and Antonio Osorio, May 21 2014

Keywords

Examples

			Add the numbers in the third column for a(n):
                                              13+ 1 + 1 + 1
                                              12+ 2 + 1 + 1
                                              11+ 3 + 1 + 1
                                              10+ 4 + 1 + 1
                                              9 + 5 + 1 + 1
                                              8 + 6 + 1 + 1
                                              7 + 7 + 1 + 1
                                              11+ 2 + 2 + 1
                                              10+ 3 + 2 + 1
                                              9 + 4 + 2 + 1
                                              8 + 5 + 2 + 1
                                              7 + 6 + 2 + 1
                                              9 + 3 + 3 + 1
                                              8 + 4 + 3 + 1
                                              7 + 5 + 3 + 1
                                              6 + 6 + 3 + 1
                                              7 + 4 + 4 + 1
                                              6 + 5 + 4 + 1
                                              5 + 5 + 5 + 1
                              9 + 1 + 1 + 1   10+ 2 + 2 + 2
                              8 + 2 + 1 + 1   9 + 3 + 2 + 2
                              7 + 3 + 1 + 1   8 + 4 + 2 + 2
                              6 + 4 + 1 + 1   7 + 5 + 2 + 2
                              5 + 5 + 1 + 1   6 + 6 + 2 + 2
                              7 + 2 + 2 + 1   8 + 3 + 3 + 2
                              6 + 3 + 2 + 1   7 + 4 + 3 + 2
                              5 + 4 + 2 + 1   6 + 5 + 3 + 2
                              5 + 3 + 3 + 1   6 + 4 + 4 + 2
                              4 + 4 + 3 + 1   5 + 5 + 4 + 2
               5 + 1 + 1 + 1  6 + 2 + 2 + 2   7 + 3 + 3 + 3
               4 + 2 + 1 + 1  5 + 3 + 2 + 2   6 + 4 + 3 + 3
               3 + 3 + 1 + 1  4 + 4 + 2 + 2   5 + 5 + 3 + 3
               3 + 2 + 2 + 1  4 + 3 + 3 + 2   5 + 4 + 4 + 3
1 + 1 + 1 + 1  2 + 2 + 2 + 2  3 + 3 + 3 + 3   4 + 4 + 4 + 4
    4(1)            4(2)           4(3)            4(4)       ..   4n
------------------------------------------------------------------------
     1               7              29              86        ..   a(n)

Links

Crossrefs

Cf. A238328, A238340, A238702, A239667, A241084.

Programs

  • Magma
    I:=[1,7,29,86,198,396,719,1203,1899]; [n le 9 select I[n] else 3*Self(n-1)-3*Self(n-2)+3*Self(n-3)-6*Self(n-4)+6*Self(n-5)-3*Self(n-6)+3*Self(n-7)-3*Self(n-8)+Self(n-9): n in [1..40]]; // Vincenzo Librandi, Aug 29 2015
  • Mathematica
    CoefficientList[Series[-(1 + 4x + 11x^2 + 17x^3 + 12x^4 + 9x^5 + 2x^6) / ((-1 + x)^5 (1 + x + x^2)^2), {x, 0, 50}], x]
LinearRecurrence[{3, -3, 3, -6, 6, -3, 3, -3, 1}, {1, 7, 29, 86, 198, 396, 719, 1203, 1899}, 50] (* Vincenzo Librandi, Aug 29 2015 *)

Formula

G.f.: (1 + 4*x + 11*x^2 + 17*x^3 + 12*x^4 + 9*x^5 + 2*x^6) / ((1 - x)^5*(1 + x + x^2)^2).
a(n) = A238328(n) - A239667(n) - A241084(n) - A238702(n).
a(n) = 7/27*n^4 + 35/27*n^3 + 22/9*n^2 + 59/27*n + O(1). - Ralf Stephan, May 26 2014

A243011 Sum of the three largest parts in the partitions of 4n into 4 parts.

Original entry on oeis.org

3, 34, 159, 489, 1161, 2365, 4336, 7323, 11640, 17646, 25702, 36246, 49761, 66720, 87685, 113263, 144039, 180699, 223974, 274561, 333270, 400956, 478428, 566620, 666511, 779022, 905211, 1046181, 1202965, 1376745, 1568748, 1780119, 2012164, 2266234, 2543586
Offset: 1

Views

Author

Wesley Ivan Hurt, May 28 2014

Keywords

Examples

			Add up the numbers in the first three columns for a(n):
                                             13 + 1 + 1 + 1
                                             12 + 2 + 1 + 1
                                             11 + 3 + 1 + 1
                                             10 + 4 + 1 + 1
                                              9 + 5 + 1 + 1
                                              8 + 6 + 1 + 1
                                              7 + 7 + 1 + 1
                                             11 + 2 + 2 + 1
                                             10 + 3 + 2 + 1
                                              9 + 4 + 2 + 1
                                              8 + 5 + 2 + 1
                                              7 + 6 + 2 + 1
                                              9 + 3 + 3 + 1
                                              8 + 4 + 3 + 1
                                              7 + 5 + 3 + 1
                                              6 + 6 + 3 + 1
                                              7 + 4 + 4 + 1
                                              6 + 5 + 4 + 1
                                              5 + 5 + 5 + 1
                              9 + 1 + 1 + 1  10 + 2 + 2 + 2
                              8 + 2 + 1 + 1   9 + 3 + 2 + 2
                              7 + 3 + 1 + 1   8 + 4 + 2 + 2
                              6 + 4 + 1 + 1   7 + 5 + 2 + 2
                              5 + 5 + 1 + 1   6 + 6 + 2 + 2
                              7 + 2 + 2 + 1   8 + 3 + 3 + 2
                              6 + 3 + 2 + 1   7 + 4 + 3 + 2
                              5 + 4 + 2 + 1   6 + 5 + 3 + 2
                              5 + 3 + 3 + 1   6 + 4 + 4 + 2
                              4 + 4 + 3 + 1   5 + 5 + 4 + 2
               5 + 1 + 1 + 1  6 + 2 + 2 + 2   7 + 3 + 3 + 3
               4 + 2 + 1 + 1  5 + 3 + 2 + 2   6 + 4 + 3 + 3
               3 + 3 + 1 + 1  4 + 4 + 2 + 2   5 + 5 + 3 + 3
               3 + 2 + 2 + 1  4 + 3 + 3 + 2   5 + 4 + 4 + 3
1 + 1 + 1 + 1  2 + 2 + 2 + 2  3 + 3 + 3 + 3   4 + 4 + 4 + 4
    4(1)            4(2)           4(3)            4(4)       ..   4n
------------------------------------------------------------------------
     3               34            159             489        ..   a(n)

Links

Crossrefs

Cf. A238328, A238340, A238702, A239667, A241084.

Programs

  • Mathematica
    a[1] = 4; a[n_] := (n/(n - 1)) a[n - 1] + 4 n*Sum[(Floor[(4 n - 2 - i)/2] - i) (Floor[(Sign[(Floor[(4 n - 2 - i)/2] - i)] + 2)/2]), {i, 0, 2 n}]; Table[a[n] - Sum[a[i]/i, {i, n}]/4, {n, 30}]
  • PARI
    Vec(-x*(16*x^6+58*x^5+87*x^4+105*x^3+66*x^2+25*x+3)/((x-1)^5*(x^2+x+1)^2) + O(x^100)) \\ Colin Barker, Sep 22 2014

Formula

a(n) = A238328(n) - A238702(n).
a(n) = A239667(n) + A241084(n) + A242727(n).
a(n) = 4n * A238340(n) - Sum_{i=1..n} A238340(i).
a(n) = (4n-1) * A238702(n) - 4n * A238702(n-1), n > 1.
a(n) = A238328(n) - (1/4) * Sum_{i=1..n} A238328(i)/i.
G.f.: -x*(16*x^6+58*x^5+87*x^4+105*x^3+66*x^2+25*x+3) / ((x-1)^5*(x^2+x+1)^2). - Colin Barker, Sep 22 2014
a(n) = 3*a(n-1) - 3*a(n-2) + 3*a(n-3) - 6*a(n-4) + 6*a(n-5) - 3*a(n-6) + 3*a(n-7) - 3*a(n-8) + a(n-9). - Wesley Ivan Hurt, Jun 20 2024
Showing 1-2 of 2 results.

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