A238328 Sum of all the parts in the partitions of 4n into 4 parts.
4, 40, 180, 544, 1280, 2592, 4732, 7968, 12636, 19120, 27808, 39168, 53716, 71960, 94500, 121984, 155040, 194400, 240844, 295120, 358092, 430672, 513728, 608256, 715300, 835848, 971028, 1122016, 1289920, 1476000, 1681564, 1907840, 2156220, 2428144, 2724960
Offset: 1
Examples
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
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4 40 180 544 .. a(n)
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..1000
- Antonio Osorio, A Sequential Allocation Problem: The Asymptotic Distribution of Resources, Munich Personal RePEc Archive, 2014.
- Index entries for sequences related to partitions
- Index entries for linear recurrences with constant coefficients, signature (3,-3,3,-6,6,-3,3,-3,1).
Programs
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Magma
I:=[4, 40, 180, 544, 1280, 2592, 4732, 7968, 12636]; [n le 9 select I[n] else 3*Self(n-1)-3*Self(n-2)+3*Self(n-3)-6*Self(n-4)+6*Self(n-5)-3*Self(n-6)+3*Self(n-7)-3*Self(n-8)+Self(n-9): n in [1..45]]; // Vincenzo Librandi, Aug 29 2015
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Mathematica
CoefficientList[Series[4*(4*x^6 + 15*x^5 + 23*x^4 + 28*x^3 + 18*x^2 + 7*x + 1)/((1 - x)^5*(x^2 + x + 1)^2), {x, 0, 50}], x] (* Wesley Ivan Hurt, Jun 27 2014 *) LinearRecurrence[{3, -3, 3, -6, 6, -3, 3, -3, 1}, {4, 40, 180, 544, 1280, 2592, 4732, 7968, 12636}, 50] (* Vincenzo Librandi, Aug 29 2015 *) -
PARI
Vec(-4*x*(4*x^6+15*x^5+23*x^4+28*x^3+18*x^2+7*x+1)/((x-1)^5*(x^2+x+1)^2) + O(x^100)) \\ Colin Barker, Mar 24 2014
Formula
Recurrence: a(1) = 4, with a(n) = (n/(n-1))*a(n-1) + 4n*Sum_{i=0..2n} (floor((4n-2-i)/2)-i) * floor((sign(floor((4n-2-i)/2)-i)+2)/2), n > 1.
G.f.: 4*x*(4*x^6+15*x^5+23*x^4+28*x^3+18*x^2+7*x+1) / ((1-x)^5*(x^2+x+1)^2). - Colin Barker, Mar 10 2014
a(n) = 16/9*n^4 + 4/3*n^3 + O(n). - Ralf Stephan, May 29 2014
a(n) = 4n * A238340(n). - Wesley Ivan Hurt, May 29 2014
E.g.f.: 4*exp(-x/2)*(3*exp(3*x/2)*(8 + x*(37 + x*(27 + 4*x))) + 3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2))/27. - Stefano Spezia, Feb 09 2023
a(n) = 3*a(n-1) - 3*a(n-2) + 3*a(n-3) - 6*a(n-4) + 6*a(n-5) - 3*a(n-6) + 3*a(n-7) - 3*a(n-8) + a(n-9). - Wesley Ivan Hurt, Jun 19 2024