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 21-30 of 30 results.

A256317 Number of partitions of 4n into exactly 6 parts.

Original entry on oeis.org

0, 0, 2, 11, 35, 90, 199, 391, 709, 1206, 1945, 3009, 4494, 6510, 9192, 12692, 17180, 22856, 29941, 38677, 49342, 62239, 77695, 96079, 117788, 143247, 172929, 207338, 247010, 292534, 344534, 403670, 470660, 546261, 631269, 726544, 832989, 951549, 1083239
Offset: 0

Views

Author

Colin Barker, Mar 23 2015

Keywords

Examples

			For n=2 the 2 partitions of 4*2 = 8 are [1,1,1,1,1,3] and [1,1,1,1,2,2].

Links

Crossrefs

Cf. A238340 (4 parts), A256316 (5 parts).

Programs

  • Mathematica
    Table[Length[IntegerPartitions[4n,{6}]],{n,0,40}] (* or *) LinearRecurrence[ {3,-3,3,-6,7,-6,6,-6,7,-6,3,-3,3,-1},{0,0,2,11,35,90,199,391,709,1206,1945,3009,4494,6510},40] (* Harvey P. Dale, Apr 12 2018 *)
  • PARI
    concat(0, vector(40, n, k=0; forpart(p=4*n, k++, , [6,6]); k))
  • PARI
    concat([0,0], Vec(x^2*(x+1)^2*(x^2+1)*(x^4+2*x^3+2*x^2+x+2) / ((x-1)^6*(x^2+x+1)^2*(x^4+x^3+x^2+x+1)) + O(x^100)))

Formula

G.f.: x^2*(x+1)^2*(x^2+1)*(x^4+2*x^3+2*x^2+x+2) / ((x-1)^6*(x^2+x+1)^2*(x^4+x^3+x^2+x+1)).

A256539 Number of partitions of 4n into at most 5 parts.

Original entry on oeis.org

1, 5, 18, 47, 101, 192, 333, 540, 831, 1226, 1747, 2418, 3266, 4319, 5608, 7166, 9027, 11229, 13811, 16814, 20282, 24260, 28796, 33940, 39744, 46262, 53550, 61667, 70673, 80631, 91606, 103664, 116875, 131310, 147042, 164147, 182702, 202787, 224484, 247877
Offset: 0

Views

Author

Colin Barker, Apr 01 2015

Keywords

Examples

			For n=2 the 18 partitions of 2*4 = 8 are [8], [1,7], [2,6], [3,5], [4,4], [1,1,6], [1,2,5], [1,3,4], [2,2,4], [2,3,3], [1,1,1,5], [1,1,2,4], [1,1,3,3], [1,2,2,3], [2,2,2,2], [1,1,1,1,4], [1,1,1,2,3] and [1,1,2,2,2].

Links

Crossrefs

Cf. A001401, A238340 (4 parts), A256540 (6 parts).

Programs

  • PARI
    concat(1, vector(40, n, k=0; forpart(p=4*n, k++, , [1,5]); k))
  • PARI
    Vec(-(x^7+4*x^6+5*x^5+7*x^4+6*x^3+6*x^2+2*x+1) / ((x-1)^5*(x^2+x+1)*(x^4+x^3+x^2+x+1)) + O(x^100))

Formula

G.f.: -(x^7+4*x^6+5*x^5+7*x^4+6*x^3+6*x^2+2*x+1) / ((x-1)^5*(x^2+x+1)*(x^4+x^3+x^2+x+1)).
a(n) = A001401(4n). - Alois P. Heinz, Apr 01 2015

A256540 Number of partitions of 4n into at most 6 parts.

Original entry on oeis.org

1, 5, 20, 58, 136, 282, 532, 931, 1540, 2432, 3692, 5427, 7760, 10829, 14800, 19858, 26207, 34085, 43752, 55491, 69624, 86499, 106491, 130019, 157532, 189509, 226479, 269005, 317683, 373165, 436140, 507334, 587535, 677571, 778311, 890691, 1015691, 1154336
Offset: 0

Views

Author

Colin Barker, Apr 01 2015

Keywords

Examples

			For n=1 the 5 partitions of 1*4 = 4 are [4], [1,3], [2,2], [1,1,2] and [1,1,1,1].

Links

Crossrefs

Cf. A001402, A238340 (4 parts), A256539 (5 parts).

Programs

  • PARI
    concat(1, vector(40, n, k=0; forpart(p=4*n, k++, , [1,6]); k))
  • PARI
    Vec((3*x^8+5*x^7+11*x^6+11*x^5+13*x^4+10*x^3+8*x^2+2*x+1) / ((x-1)^6*(x^2+x+1)^2*(x^4+x^3+x^2+x+1)) + O(x^100))

Formula

G.f.: (3*x^8+5*x^7+11*x^6+11*x^5+13*x^4+10*x^3+8*x^2+2*x+1) / ((x-1)^6*(x^2+x+1)^2*(x^4+x^3+x^2+x+1)).
a(n) = A001402(4n). - Alois P. Heinz, Apr 01 2015

A239195 Sum of the next to smallest parts in the partitions of 4n into 4 parts with smallest part = 1.

Original entry on oeis.org

1, 5, 17, 42, 78, 134, 215, 315, 447, 616, 812, 1052, 1341, 1665, 2045, 2486, 2970, 3522, 4147, 4823, 5579, 6420, 7320, 8312, 9401, 10557, 11817, 13186, 14630, 16190, 17871, 19635, 21527, 23552, 25668, 27924, 30325, 32825, 35477, 38286, 41202, 44282, 47531
Offset: 1

Views

Author

Wesley Ivan Hurt, Mar 11 2014

Keywords

Examples

			For a(n) add the numbers in the third columns.
                                               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 + 1 + 1 + 1    9 + 4 + 2 + 1
                              8 + 2 + 1 + 1    8 + 5 + 2 + 1
                              7 + 3 + 1 + 1    7 + 6 + 2 + 1
                              6 + 4 + 1 + 1    9 + 3 + 3 + 1
                              5 + 5 + 1 + 1    8 + 4 + 3 + 1
                              7 + 2 + 2 + 1    7 + 5 + 3 + 1
               5 + 1 + 1 + 1  6 + 3 + 2 + 1    6 + 6 + 3 + 1
               4 + 2 + 1 + 1  5 + 4 + 2 + 1    7 + 4 + 4 + 1
               3 + 3 + 1 + 1  5 + 3 + 3 + 1    6 + 5 + 4 + 1
1 + 1 + 1 + 1  3 + 2 + 2 + 1  4 + 4 + 3 + 1    5 + 5 + 5 + 1
    4(1)            4(2)           4(3)            4(4)       ..   4n
------------------------------------------------------------------------
     1               5              17              42        ..   a(n)

Links

Crossrefs

Cf. A238328, A238340, A238702, A238705, A238706, A239056, A239057, A239059, A239186.

Programs

  • Mathematica
    b[n_] := Sum[(((4 n - 2 - i)*Floor[(4 n - 2 - i)/2] - i (4 n - 2 - i) + (i + 2) (Floor[(4 n - 2 - i)/2] - i)) - ((4 n - 2 - i)*Floor[(4 n - 2 - i)/2] - i (4 n - 2 - i)) - ((4 n - 2 - i)*Floor[(4 n - 2 - i)/2] - i (4 n - 2 - i) + (i + 2) (Floor[(4 n - 2 - i)/2] - i))/(4 n)) (Floor[(Sign[(Floor[(4 n - 2 - i)/2] - i)] + 2)/2]), {i, 0, 2 n}]; Table[b[n], {n, 50}]
LinearRecurrence[{2,-1,2,-4,2,-1,2,-1},{1,5,17,42,78,134,215,315},60] (* Harvey P. Dale, Jul 05 2025 *)
  • PARI
    Vec(x*(4*x^5+5*x^4+11*x^3+8*x^2+3*x+1)/((x-1)^4*(x^2+x+1)^2) + O(x^100)) \\ Colin Barker, Sep 22 2014

Formula

G.f.: x*(4*x^5+5*x^4+11*x^3+8*x^2+3*x+1) / ((x-1)^4*(x^2+x+1)^2). - Colin Barker, Mar 12 2014
a(n) = 2*a(n-1)-a(n-2)+2*a(n-3)-4*a(n-4)+2*a(n-5)-a(n-6)+2*a(n-7)-a(n-8). - Wesley Ivan Hurt, Jul 08 2025

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

A262672 Expansion of (3-x-x^3) / ((x-1)^2*(1+x+x^2+x^3)).

Original entry on oeis.org

3, 2, 2, 1, 4, 3, 3, 2, 5, 4, 4, 3, 6, 5, 5, 4, 7, 6, 6, 5, 8, 7, 7, 6, 9, 8, 8, 7, 10, 9, 9, 8, 11, 10, 10, 9, 12, 11, 11, 10, 13, 12, 12, 11, 14, 13, 13, 12, 15, 14, 14, 13, 16, 15, 15, 14, 17, 16, 16, 15, 18, 17, 17, 16, 19, 18, 18, 17, 20, 19, 19, 18, 21
Offset: 0

Views

Author

Wesley Ivan Hurt, Sep 26 2015

Keywords

Comments

From Altug Alkan, Sep 29 2015: (Start)
Sequence can be defined as a composition of arithmetic sequences and it can be generated by the equations: a(4k) = k+3, a(4k+3) = k+1, a(4k+1) = a(4k+2) = k+2, with k >= 0.
Obviously, minimum value of sequence is a(3), which is also unique value that has no repetition in sequence. (End)

Examples

			G.f. = 3 + 2*x + 2*x^2 + x^3 + 4*x^4 + 3*x^5 + 3*x^6 + 2*x^7 + 5*x^8 + 4*x^9 + ...

Links

Crossrefs

Row 2 of the example in A238340 (read from bottom).
Cf. A002265, A028242, A118402.

Programs

  • Magma
    [(2*n+13+5*(-1)^n-6*(-1)^((2*n+3+(-1)^n) div 4))/8 : n in [0..100]];
  • Magma
    &cat[[3+n,2+n,2+n,1+n]: n in [0..20]]; // Bruno Berselli, Sep 29 2015
  • Magma
    I:=[3,2,2,1,4]; [n le 5 select I[n] else Self(n-1) + Self(n-4) - Self(n-5): n in [1..100]]; // Vincenzo Librandi, Sep 29 2015
  • Maple
    A262672:=n->(2*n+13+5*(-1)^n-6*(-1)^((2*n+3+(-1)^n)/4))/8: seq(A262672(n), n=0..100);
  • Mathematica
    Table[(2n + 13 + 5 (-1)^n - 6 (-1)^((2n + 3 + (-1)^n)/4))/8, {n, 0, 100}]
LinearRecurrence[{1, 0, 0, 1, -1}, {3, 2, 2, 1, 4}, 100] (* Vincenzo Librandi, Sep 29 2015 *)
CoefficientList[Series[(3-x-x^3)/((x-1)^2(1+x+x^2+x^3)),{x,0,100}],x] (* Harvey P. Dale, May 26 2023 *)
  • PARI
    a(n) = (2*n+13+5*(-1)^n-6*(-1)^((2*n+3+(-1)^n)/4))/8;
vector(80, n, a(n-1)) \\ Altug Alkan, Sep 29 2015
  • PARI
    {a(n) = my(k = n\4); [ 3, 2, 2, 1][n%4 + 1] + k}; /* Michael Somos, Oct 02 2015 */

Formula

G.f.: (3-x-x^3) / ((x-1)^2*(1+x+x^2+x^3)).
a(n) = a(n-1) + a(n-4) - a(n-5), for n>4
a(n) = ( 2*n + 13+5*(-1)^n - 6*(-1)^((2*n + 3 + (-1)^n)/4) )/8.
a(n) = 3*A002265(n+4) - A002265(n+3) - A002265(n+1).
a(2n) = A028242(n+4), a(2n+1) = A028242(n+2).
4*a(n) = |A118402(n+8)| + 3*i^(n*(n-1)), where i=sqrt(-1). [Bruno Berselli, Sep 29 2015]
E.g.f.: (1/8)*(2*x*exp(x) + 5*exp(-x) + 13*exp(x) + 6*sin(x) + 6*cos(x)). - G. C. Greubel, Sep 29 2015
a(n) = -a(-13 - n) for all n in Z. - Michael Somos, Oct 02 2015

Extensions

Edited by Bruno Berselli, Sep 30 2015

A240711 Sum of the largest parts in the partitions of 4n into 4 parts with smallest part = 1.

Original entry on oeis.org

1, 15, 62, 163, 333, 596, 973, 1475, 2130, 2959, 3969, 5192, 6649, 8343, 10310, 12571, 15125, 18012, 21253, 24843, 28826, 33223, 38025, 43280, 49009, 55199, 61902, 69139, 76893, 85220, 94141, 103635, 113762, 124543, 135953, 148056, 160873, 174375, 188630
Offset: 1

Views

Author

Wesley Ivan Hurt, Apr 10 2014

Keywords

Examples

			For a(n) add the parts in the first columns.
                                              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 + 1 + 1 + 1    9 + 4 + 2 + 1
                              8 + 2 + 1 + 1    8 + 5 + 2 + 1
                              7 + 3 + 1 + 1    7 + 6 + 2 + 1
                              6 + 4 + 1 + 1    9 + 3 + 3 + 1
                              5 + 5 + 1 + 1    8 + 4 + 3 + 1
                              7 + 2 + 2 + 1    7 + 5 + 3 + 1
               5 + 1 + 1 + 1  6 + 3 + 2 + 1    6 + 6 + 3 + 1
               4 + 2 + 1 + 1  5 + 4 + 2 + 1    7 + 4 + 4 + 1
               3 + 3 + 1 + 1  5 + 3 + 3 + 1    6 + 5 + 4 + 1
1 + 1 + 1 + 1  3 + 2 + 2 + 1  4 + 4 + 3 + 1    5 + 5 + 5 + 1
    4(1)            4(2)           4(3)            4(4)       ..   4n
------------------------------------------------------------------------
     1               15             62             163       ..   a(n)

Links

Crossrefs

Cf. A238328, A238340, A238702, A238705, A238706, A239056, A239057, A239059, A240707.

Programs

  • Mathematica
    b[n_] := Sum[((4 n - 2 - i)*Floor[(4 n - 2 - i)/2] - i (4 n - 2 - i)) (Floor[(Sign[(Floor[(4 n - 2 - i)/2] - i)] + 2)/2]), {i, 0, 2 n}]; c[1] = 1; c[n_] := Sum[Sum[i (Floor[(Sign[(Floor[(4 n - 2 - j)/2] - j)] + 2)/2]), {i, j + 1, Floor[(4 n - 2 - j)/2]}], {j, 0, 2 n}]; Table[b[n] - c[n], {n, 50}]
  • PARI
    Vec(x*(7*x^6+27*x^5+43*x^4+52*x^3+33*x^2+13*x+1)/((x-1)^4*(x^2+x+1)^2) + O(x^100)) \\ Colin Barker, Apr 11 2014

Formula

G.f.: x*(7*x^6+27*x^5+43*x^4+52*x^3+33*x^2+13*x+1) / ((x-1)^4*(x^2+x+1)^2). - Colin Barker, Apr 11 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

A308870 Sum of the fourth largest parts in the partitions of n into 6 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 6, 9, 15, 20, 31, 42, 61, 80, 112, 143, 191, 243, 316, 393, 501, 613, 767, 930, 1141, 1367, 1659, 1967, 2354, 2769, 3279, 3824, 4491, 5196, 6047, 6956, 8031, 9181, 10536, 11971, 13647, 15434, 17497, 19690, 22211, 24880, 27929
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 29 2019

Keywords

Links

Crossrefs

Cf. A238340, A308867, A308868, A308869, A306671, A308872, A308873.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[k, {i, j, Floor[(n - j - k - l - m)/2]}], {j, k, Floor[(n - k - l - m)/3]}], {k, l, Floor[(n - l - m)/4]}], {l, m, Floor[(n - m)/5]}], {m, Floor[n/6]}], {n, 0, 50}]
Table[Total[IntegerPartitions[n,{6}][[;;,4]]],{n,0,50}] (* Harvey P. Dale, Jul 30 2024 *)

Formula

a(n) = Sum_{m=1..floor(n/6)} Sum_{l=m..floor((n-m)/5)} Sum_{k=l..floor((n-l-m)/4)} Sum_{j=k..floor((n-k-l-m)/3)} Sum_{i=j..floor((n-j-k-l-m)/2)} k.
a(n) = A308867(n) - A308868(n) - A308869(n) - A306671(n) - A308872(n) - A308873(n).

A308871 Sum of the third largest parts in the partitions of n into 6 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 7, 11, 19, 26, 40, 57, 81, 109, 153, 198, 264, 342, 442, 556, 710, 875, 1093, 1338, 1638, 1975, 2398, 2855, 3416, 4040, 4779, 5595, 6573, 7627, 8875, 10244, 11822, 13549, 15553, 17707, 20187, 22883, 25935, 29239, 32991, 37010
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 29 2019

Keywords

Links

Crossrefs

Cf. A238340, A308867, A308868, A308869, A306670, A308872, A308873.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[j, {i, j, Floor[(n - j - k - l - m)/2]}], {j, k, Floor[(n - k - l - m)/3]}], {k, l, Floor[(n - l - m)/4]}], {l, m, Floor[(n - m)/5]}], {m, Floor[n/6]}], {n, 0, 50}]

Formula

a(n) = Sum_{m=1..floor(n/6)} Sum_{l=m..floor((n-m)/5)} Sum_{k=l..floor((n-l-m)/4)} Sum_{j=k..floor((n-k-l-m)/3)} Sum_{i=j..floor((n-j-k-l-m)/2)} j.
a(n) = A308867(n) - A308868(n) - A308869(n) - A306670(n) - A308872(n) - A308873(n).
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