A235988 Sum of the partition parts of 3n into 3 parts.
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
Offset: 1
Examples
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
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3 18 63 144 285 .. a(n)
- _Wesley Ivan Hurt_, Sep 07 2019
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).
Programs
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Magma
[3*n^3-3*n*Floor(n^2/4): n in [1..100]]; // Wesley Ivan Hurt, Nov 01 2015
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Magma
[3*n*(1-(-1)^n+6*n^2)/8: n in [1..40]]; // Vincenzo Librandi, Nov 18 2015
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Maple
A235988:=n->3*n^3 - 3*n*floor(n^2/4); seq(A235988(n), n=1..100);
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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 *) -
PARI
a(n)=3*n^3 - n^2\4*3*n \\ Charles R Greathouse IV, Oct 07 2015
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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
Formula
a(n) = 3*n^3 - 3*n*floor(n^2/4).
a(n) = 3n * A077043(n).
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
E.g.f.: 3*x*((4 + 9*x + 3*x^2)*cosh(x) + 3*(1 + 3*x + x^2)*sinh(x))/4. - Stefano Spezia, Feb 09 2023
Extensions
a(165) in b-file corrected by Andrew Howroyd, Feb 21 2018