OFFSET
1,2
LINKS
A. Osorio, A Sequential Allocation Problem: The Asymptotic Distribution of Resources, Munich Personal RePEc Archive, 2014.
Index entries for linear recurrences with constant coefficients, signature (3,-3,3,-6,6,-3,3,-3,1).
FORMULA
G.f.: -x*(5*x^6+17*x^5+25*x^4+30*x^3+19*x^2+7*x+1) / ((x-1)^5*(x^2+x+1)^2). - Colin Barker, Apr 16 2014
Recurrence: 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)) for n>1. Then a(1) = 1, with a(n) = a(n-1) + b(n-1)/(4n-4) + Sum_{j=0..2n} (Sum_{i=j+1..floor((4n-2-j)/2)} i * (floor((sign((floor((4n-2-j)/2)-j))+ 2)/2)) ), for n>1. - Wesley Ivan Hurt, Jun 27 2014
EXAMPLE
For a(n) add the numbers in the second columns.
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
------------------------------------------------------------------------
1 10 46 141 .. a(n)
MATHEMATICA
CoefficientList[Series[-(5*x^6 + 17*x^5 + 25*x^4 + 30*x^3 + 19*x^2 + 7*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, 10, 46, 141, 334, 680, 1247, 2106, 3348}, 50] (* Vincenzo Librandi, Aug 29 2015 *)
Table[Total[IntegerPartitions[4 n, {4}][[;; , 2]]], {n, 40}] (* Harvey P. Dale, Aug 17 2024 *)
PROG
(PARI) Vec(-x*(5*x^6+17*x^5+25*x^4+30*x^3+19*x^2+7*x+1)/((x-1)^5*(x^2+x+1)^2) + O(x^100)) \\ Colin Barker, Apr 16 2014
(Magma) I:=[1, 10, 46, 141, 334, 680, 1247, 2106, 3348]; [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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt and Antonio Osorio, Apr 15 2014
STATUS
approved