A236364 Sum of all the middle parts in the partitions of 3n into 3 parts.

1, 5, 18, 40, 80, 135, 217, 320, 459, 625, 836, 1080, 1378, 1715, 2115, 2560, 3077, 3645, 4294, 5000, 5796, 6655, 7613, 8640, 9775, 10985, 12312, 13720, 15254, 16875, 18631, 20480, 22473, 24565, 26810, 29160, 31672, 34295, 37089, 40000, 43091, 46305, 49708

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) = A000330(n) + Sum_{i=1..floor((n-1)/2)} (n + i)*(n - 2i).

a(n) = n*(n+1)*(2*n+1)/6 - floor((n-1)/2) * (4*floor((n-1)/2)^2 + 3*(n+2)*floor((n-1)/2) - 6*n^2 + 3*n + 2)/6.

G.f.: x*(x^4+3*x^3+7*x^2+3*x+1)/((x-1)^4*(x+1)^2). - Joerg Arndt, Jan 23 2014

a(n) = (n*(3-3*(-1)^n+10*n^2))/16. - Colin Barker, Jan 23 2014

a(n) = (n^3 + ceiling(n/2)^3 + floor(n/2)^3)/2. - Wesley Ivan Hurt, Apr 15 2016

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 second 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           5          18           40          80      ..   a(n)

Maple

A236364:=n->n*(n+1)*(2*n+1)/6 - floor((n-1)/2) * (4*floor((n-1)/2)^2 + (3*n+6)*floor((n-1)/2) - 6*n^2 + 3*n + 2)/6; seq(A236364(n), n=1..100);

Mathematica

Table[Sum[i^2, {i, n}] + Sum[(n + i) (n - 2 i), {i, Floor[(n - 1)/2]}], {n, 100}]
CoefficientList[Series[(x^4 + 3 x^3 + 7 x^2 + 3 x + 1)/ ((x - 1)^4 (x + 1)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Feb 18 2014 *)

Prog

(PARI) Vec(x*(x^4+3*x^3+7*x^2+3*x+1)/((x-1)^4*(x+1)^2) + O(x^100)) \\ Colin Barker, Jan 23 2014

Crossrefs

Cf. A000330, A019298, A235988.

Sequence in context: A007742 A225272 A276819 * A352368 A000338 A212343

Adjacent sequences: A236361 A236362 A236363 * A236365 A236366 A236367

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Jan 23 2014

Extensions

Name clarified by Wesley Ivan Hurt, Apr 16 2016

Status

approved