A239059 Sum of the two smallest parts from the partitions of 4n into 4 parts with smallest part = 1.

2, 9, 27, 61, 108, 178, 276, 395, 549, 743, 966, 1236, 1558, 1917, 2335, 2817, 3344, 3942, 4616, 5343, 6153, 7051, 8010, 9064, 10218, 11441, 12771, 14213, 15732, 17370, 19132, 20979, 22957, 25071, 27278, 29628, 32126, 34725, 37479, 40393, 43416, 46606, 49968

Offset

1,1

Links

Iain Fox, Table of n, a(n) for n = 1..10000 (first 200 terms from Vincenzo Librandi)

Index entries for sequences related to partitions

Index entries for linear recurrences with constant coefficients, signature (2,-1,2,-4,2,-1,2,-1).

Formula

G.f.: -x*(x^2+x+2)*(2*x^4-3*x^3-4*x^2-2*x-1) / ((x-1)^4*(x^2+x+1)^2). - Colin Barker, Mar 10 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, Jun 20 2024

Example

For a(n) add the smallest two parts in the partitions with smallest part equal to 1.
                                             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
------------------------------------------------------------------------
     2               9              27              61        ..   a(n)

Mathematica

b[n_] := Sum[((i + 2) (Floor[(4 n - 2 - i)/2] - i)) (Floor[(Sign[(Floor[(4 n - 2 - i)/2] - i)] + 2)/2]), {i, 0, 2 n}]; Table[b[n], {n, 50}]

Prog

(PARI) Vec(-x*(x^2+x+2)*(2*x^4-3*x^3-4*x^2-2*x-1)/((x-1)^4*(x^2+x+1)^2) + O(x^100)) \\ Colin Barker, Sep 22 2014

Crossrefs

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

Sequence in context: A182325 A008910 A373603 * A153977 A051746 A277240

Adjacent sequences: A239056 A239057 A239058 * A239060 A239061 A239062

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Mar 09 2014

Status

approved