A238705 Number of partitions of 4n into 4 parts with smallest part = 1.
1, 4, 10, 19, 30, 44, 61, 80, 102, 127, 154, 184, 217, 252, 290, 331, 374, 420, 469, 520, 574, 631, 690, 752, 817, 884, 954, 1027, 1102, 1180, 1261, 1344, 1430, 1519, 1610, 1704, 1801, 1900, 2002, 2107, 2214, 2324, 2437, 2552, 2670, 2791, 2914, 3040, 3169
Offset: 1
Examples
Count the 1's 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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1 4 10 19 .. a(n)
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..200
- 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 (2,-1,1,-2,1).
Programs
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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); Table[b[n] - b[n - 1], {n, 50}] LinearRecurrence[{2,-1,1,-2,1},{1,4,10,19,30},50] (* Harvey P. Dale, Jun 13 2015 *) Table[Count[IntegerPartitions[4 n,{4}],?(#[[-1]]==1&)],{n,50}] (* _Harvey P. Dale, Dec 29 2021 *) -
PARI
Vec(-x*(x+1)*(2*x^2+x+1)/((x-1)^3*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Sep 22 2014
Formula
G.f.: -x*(x+1)*(2*x^2+x+1) / ((x-1)^3*(x^2+x+1)). - Colin Barker, Mar 10 2014
a(n) = 2*a(n-1)-a(n-2)+a(n-3)-2*a(n-4)+a(n-5). - Wesley Ivan Hurt, Nov 18 2021
Comments