A238706 Sum of the smallest parts of the partitions of 4n into 4 parts with smallest part greater than 1.
0, 2, 11, 36, 89, 183, 335, 565, 894, 1347, 1952, 2738, 3738, 4988, 6525, 8390, 10627, 13281, 16401, 20039, 24248, 29085, 34610, 40884, 47972, 55942, 64863, 74808, 85853, 98075, 111555, 126377, 142626, 160391, 179764, 200838, 223710, 248480, 275249, 304122
Offset: 1
Examples
Add the numbers > 1 in the last 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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0 2 11 36 .. a(n)
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..1000
- 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 (4,-6,5,-5,6,-4,1).
Programs
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Magma
I:=[0,2,11,36,89,183,335]; [n le 7 select I[n] else 4*Self(n-1)-6*Self(n-2)+5*Self(n-3)-5*Self(n-4)+6*Self(n-5)-4*Self(n-6)+Self(n-7): n in [1..40]]; // Vincenzo Librandi, Mar 24 2014
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Mathematica
a[1] = 4; a[n_] := (n/(n - 1))*a[n - 1] + 4 n*Sum[(Floor[(4 n - 2 - i)/2] - i)*(Floor[(Sign[(Floor[(4 n - 2 - i)/2] - i)] + 2)/2]), {i, 0, 2 n}]; b[n_] := a[n]/(4 n); b[0] = 0; c[1] = 1; c[n_] := b[n] + c[n - 1]; Table[c[n] - (b[n] - b[n - 1]), {n, 50}] CoefficientList[Series[x (x - 2) (x + 1) (2 x^2 + x + 1)/((x - 1)^5 (x^2 + x + 1)), {x, 0, 40}],x] (* Vincenzo Librandi, Mar 24 2014 *) Table[Total[Select[IntegerPartitions[4n,{4}],#[[-1]]>1&][[All,-1]]],{n,40}] (* or *) LinearRecurrence[{4,-6,5,-5,6,-4,1},{0,2,11,36,89,183,335},40] (* Harvey P. Dale, Jan 06 2023 *) -
PARI
concat(0, Vec(x^2*(x-2)*(x+1)*(2*x^2+x+1)/((x-1)^5*(x^2+x+1)) + O(x^100))) \\ Colin Barker, Mar 23 2014
Formula
G.f.: x^2*(x-2)*(x+1)*(2*x^2+x+1) / ((x-1)^5*(x^2+x+1)). - Colin Barker, Mar 23 2014
a(n) = 4*a(n-1) - 6*a(n-2) + 5*a(n-3) - 5*a(n-4) + 6*a(n-5) - 4*a(n-6) + a(n-7) for n > 7. - Wesley Ivan Hurt, Oct 07 2017