A239056 Sum of the parts in the partitions of 4n into 4 parts with smallest part = 1.
4, 32, 120, 304, 600, 1056, 1708, 2560, 3672, 5080, 6776, 8832, 11284, 14112, 17400, 21184, 25432, 30240, 35644, 41600, 48216, 55528, 63480, 72192, 81700, 91936, 103032, 115024, 127832, 141600, 156364, 172032, 188760, 206584, 225400, 245376, 266548, 288800
Offset: 1
Examples
For a(n) add the parts in the partitions of 4n with smallest part = 1.
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 + 1 + 1 + 1 9 + 4 + 2 + 1
8 + 2 + 1 + 1 8 + 5 + 2 + 1
7 + 3 + 1 + 1 7 + 6 + 2 + 1
6 + 4 + 1 + 1 9 + 3 + 3 + 1
5 + 5 + 1 + 1 8 + 4 + 3 + 1
7 + 2 + 2 + 1 7 + 5 + 3 + 1
5 + 1 + 1 + 1 6 + 3 + 2 + 1 6 + 6 + 3 + 1
4 + 2 + 1 + 1 5 + 4 + 2 + 1 7 + 4 + 4 + 1
3 + 3 + 1 + 1 5 + 3 + 3 + 1 6 + 5 + 4 + 1
1 + 1 + 1 + 1 3 + 2 + 2 + 1 4 + 4 + 3 + 1 5 + 5 + 5 + 1
4(1) 4(2) 4(3) 4(4) .. 4n
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4 32 120 304 .. a(n)
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..200
- Index entries for sequences related to partitions
- Index entries for linear recurrences with constant coefficients, signature (2,-1,2,-4,2,-1,2,-1).
Programs
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Mathematica
b[n_] := Sum[((4 n - 2 - i)*Floor[(4 n - 2 - i)/2] - i (4 n - 2 - i) + (i + 2) (Floor[(4 n - 2 - i)/2] - i)) (Floor[(Sign[(Floor[(4 n - 2 - i)/2] - i)] + 2)/2]), {i, 0, 2 n}]; Table[b[n], {n, 50}] LinearRecurrence[{2,-1,2,-4,2,-1,2,-1},{4,32,120,304,600,1056,1708,2560},40] (* Harvey P. Dale, Oct 18 2018 *) -
PARI
Vec(4*x*(2*x^6+10*x^5+16*x^4+22*x^3+15*x^2+6*x+1)/((x-1)^4*(x^2+x+1)^2) + O(x^100)) \\ Colin Barker, Sep 22 2014
Formula
G.f.: 4*x*(2*x^6+10*x^5+16*x^4+22*x^3+15*x^2+6*x+1) / ((x-1)^4*(x^2+x+1)^2). - Colin Barker, Mar 10 2014
a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3) - 4*a(n-4) + 2*a(n-5) - a(n-6) + 2*a(n-7) - a(n-8). - Wesley Ivan Hurt, Jun 20 2024
Comments