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A235988 Sum of the partition parts of 3n into 3 parts. 10
3, 18, 63, 144, 285, 486, 777, 1152, 1647, 2250, 3003, 3888, 4953, 6174, 7605, 9216, 11067, 13122, 15447, 18000, 20853, 23958, 27393, 31104, 35175, 39546, 44307, 49392, 54897, 60750, 67053, 73728, 80883, 88434, 96495, 104976, 113997, 123462, 133497, 144000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

LINKS

G. C. Greubel, 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) = 3*n^3 - 3*n*floor(n^2/4).

a(n) = 3n * A077043(n).

a(n) = a(n-1) + 3*A077043(n-1) + A001651(n) + A093353(3n-2).

From Colin Barker, Jan 18 2014: (Start)

a(n) = (3*n*(1-(-1)^n+6*n^2))/8.

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

a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n > 6. - Wesley Ivan Hurt, Nov 15 2015

EXAMPLE

a(2) = 18; 3(2) = 6 has 3 partitions into 3 parts: (4, 1, 1), (3, 2, 1), and (2, 2, 2). The sum of the parts is 18.

Figure 1: The partitions of 3n into 3 parts for n = 1, 2, 3, ...

                                               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

------------------------------------------------------------------------

    3           18          63         144         285      ..   a(n)

- Wesley Ivan Hurt, Sep 07 2019

MAPLE

A235988:=n->3*n^3 - 3*n*floor(n^2/4); seq(A235988(n), n=1..100);

MATHEMATICA

Table[3 n^3 - 3 n*Floor[n^2/4], {n, 100}] (* or *) CoefficientList[ Series[3*x*(x^4 + 4*x^3 + 8*x^2 + 4*x + 1)/((x - 1)^4*(x + 1)^2), {x, 0, 30}], x]

LinearRecurrence[{2, 1, -4, 1, 2, -1}, {3, 18, 63, 144, 285, 486}, 40] (* Harvey P. Dale, May 17 2018 *)

PROG

(PARI) a(n)=3*n^3 - n^2\4*3*n \\ Charles R Greathouse IV, Oct 07 2015

(PARI) x='x+O('x^50); Vec(3*x*(x^4+4*x^3+8*x^2+4*x+1)/((x-1)^4*(x+1)^2)) \\ Altug Alkan, Nov 01 2015

(MAGMA) [3*n^3-3*n*Floor(n^2/4): n in [1..100]]; // Wesley Ivan Hurt, Nov 01 2015

(MAGMA) [3*n*(1-(-1)^n+6*n^2)/8: n in [1..40]]; // Vincenzo Librandi, Nov 18 2015

CROSSREFS

Cf. A001651, A077043, A093353.

Sequence in context: A299031 A210366 A000648 * A253942 A317404 A110689

Adjacent sequences:  A235985 A235986 A235987 * A235989 A235990 A235991

KEYWORD

nonn,easy

AUTHOR

Wesley Ivan Hurt, Jan 17 2014

EXTENSIONS

a(165) in b-file corrected by Andrew Howroyd, Feb 21 2018

STATUS

approved

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Last modified October 23 20:17 EDT 2019. Contains 328373 sequences. (Running on oeis4.)