A236370 Sum of the largest parts in the partitions of 3n into 3 parts.

1, 9, 34, 81, 163, 282, 454, 678, 973, 1335, 1786, 2319, 2959, 3696, 4558, 5532, 6649, 7893, 9298, 10845, 12571, 14454, 16534, 18786, 21253, 23907, 26794, 29883, 33223, 36780, 40606, 44664, 49009, 53601, 58498, 63657, 69139, 74898, 80998, 87390, 94141

Offset

1,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for sequences related to partitions

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

Formula

a(n) = 3n * (n^2 - floor(n^2/4)) - Sum_{i=1..n} (2*i^2 - floor(i^2/4)) - Sum_{i=1..floor((n-1)/2)} (n + i) * (n - 2i).

From Colin Barker, Jan 24 2014: (Start)

a(n) = (-1+(-1)^n-(1+3*(-1)^n)*n-6*n^2+22*n^3)/16.

G.f.: x*(2*x^4+8*x^3+15*x^2+7*x+1) / ((x-1)^4*(x+1)^2). (End)

a(n) = Sum_{j=0..n-2} (Sum_{i=n+1+floor(j/2)-floor(1/j+1)..n+2*(j+1)} i), n > 1. - Wesley Ivan Hurt, Feb 10 2014

a(n) = 2*a(n-1)+a(n-2)-4*a(n-3)+a(n-4)+2*a(n-5)-a(n-6). - Wesley Ivan Hurt, Nov 19 2021

Example

Add first columns for a(n)..
                                               13 + 1 + 1
                                               12 + 2 + 1
                                               11 + 3 + 1
                                               10 + 4 + 1
                                                9 + 5 + 1
                                                8 + 6 + 1
                                                7 + 7 + 1
                                   10 + 1 + 1  11 + 2 + 2
                                    9 + 2 + 1  10 + 3 + 2
                                    8 + 3 + 1   9 + 4 + 2
                                    7 + 4 + 1   8 + 5 + 2
                                    6 + 5 + 1   7 + 6 + 2
                        7 + 1 + 1   8 + 2 + 2   9 + 3 + 3
                        6 + 2 + 1   7 + 3 + 2   8 + 4 + 3
                        5 + 3 + 1   6 + 4 + 2   7 + 5 + 3
                        4 + 4 + 1   5 + 5 + 2   6 + 6 + 3
            4 + 1 + 1   5 + 2 + 2   6 + 3 + 3   7 + 4 + 4
            3 + 2 + 1   4 + 3 + 2   5 + 4 + 3   6 + 5 + 4
1 + 1 + 1   2 + 2 + 2   3 + 3 + 3   4 + 4 + 4   5 + 5 + 5
   3(1)        3(2)        3(3)        3(4)        3(5)     ..   3n
---------------------------------------------------------------------
    1           9          34           81          163      ..  a(n)

Mathematica

Table[3 n (n^2 - Floor[n^2/4]) - Sum[2 i^2 - Floor[i^2/4], {i, n}] -
  Sum[(n + i) (n - 2 i), {i, Floor[(n - 1)/2]}], {n, 100}]
LinearRecurrence[{2, 1, -4, 1, 2, -1}, {1, 9, 34, 81, 163, 282}, 50] (* Harvey P. Dale, Nov 11 2017 *)

Prog

(PARI) Vec(x*(2*x^4+8*x^3+15*x^2+7*x+1)/((x-1)^4*(x+1)^2) + O(x^100)) \\ Colin Barker, Jan 24 2014

Crossrefs

Cf. A019298, A235988, A236364, A236758, A236762, A237264.

Sequence in context: A020163 A262959 A139275 * A273744 A133547 A100179

Adjacent sequences: A236367 A236368 A236369 * A236371 A236372 A236373

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Jan 23 2014

Status

approved