A239667 Sum of the largest parts of the partitions of 4n into 4 parts.
1, 17, 84, 262, 629, 1289, 2370, 4014, 6393, 9703, 14150, 19974, 27439, 36815, 48410, 62556, 79587, 99879, 123832, 151844, 184359, 221845, 264764, 313628, 368973, 431325, 501264, 579394, 666305, 762645, 869086, 986282, 1114949, 1255827, 1409634, 1577154, 1759195, 1956539, 2170038, 2400568
Offset: 1
Examples
Add the numbers in the first column for a(n):
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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1 17 84 262 .. a(n)
Links
- A. 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:=[1,17,84,262,629,1289,2370,4014,6393]; [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[-(9*x^6 + 32*x^5 + 50*x^4 + 58*x^3 + 36*x^2 + 14*x + 1)/((x - 1)^5*(x^2 + x + 1)^2), {x, 0, 50}], x] (* Wesley Ivan Hurt, Jun 13 2014 *) LinearRecurrence[{3, -3, 3, -6, 6, -3, 3, -3, 1}, {1, 17, 84, 262, 629, 1289, 2370, 4014, 6393}, 50](* Vincenzo Librandi, Aug 29 2015 *) Table[Total[IntegerPartitions[4 n,{4}][[All,1]]],{n,40}] (* Harvey P. Dale, Apr 25 2020 *) -
PARI
Vec(-x*(9*x^6+32*x^5+50*x^4+58*x^3+36*x^2+14*x+1) / ((x-1)^5*(x^2+x+1)^2) + O(x^100)) \\ Colin Barker, Mar 23 2014
Formula
G.f.: -x*(9*x^6+32*x^5+50*x^4+58*x^3+36*x^2+14*x+1) / ((x-1)^5*(x^2+x+1)^2). - Colin Barker, Mar 23 2014
Let b(1) = 4, with b(n) = (n/(n-1)) * b(n-1) + 4n * Sum_{i=0..2n} (floor((4n-2-i)/2)-i) * floor((sign(floor((4n-2-i)/2)-i)+2)/2). Then a(1) = 1, with a(n) = a(n-1) + b(n-1)/(4n-4) + Sum_{i=j+1..floor((4n-2-j)/2)} ( Sum_{j=0..2n} (4n-2-i-j) * floor((sign(floor((4n-2-j)/2)-j)+2)/2) ). - Wesley Ivan Hurt, Jun 13 2014