A350043 Sum of all the parts > 1 in the partitions of n into 3 positive integer parts.
0, 2, 7, 15, 24, 36, 58, 75, 104, 138, 175, 217, 277, 328, 399, 477, 560, 650, 766, 869, 1000, 1140, 1287, 1443, 1633, 1806, 2015, 2235, 2464, 2704, 2986, 3247, 3552, 3870, 4199, 4541, 4933, 5300, 5719, 6153, 6600, 7062, 7582, 8073, 8624, 9192, 9775, 10375, 11041, 11674
Offset: 3
Examples
a(7) = 24; The partitions of 7 into 3 positive integer parts are (1,1,5), (1,2,4), (1,3,3) and (2,2,3). The sum of all the parts > 1 is then 5+2+4+3+3+2+2+3 = 24.
Links
- Winston de Greef, Table of n, a(n) for n = 3..10000
- Index entries for sequences related to partitions
- Index entries for linear recurrences with constant coefficients, signature (0,2,2,-1,-4,-1,2,2,0,-1).
Programs
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Mathematica
CoefficientList[Series[-x*(x^9 - x^8 - 3*x^7 + 5*x^5 + 6*x^4 - 6*x^3 - 11*x^2 - 7*x - 2)/((x + 1)^2*(x^2 + x + 1)^2*(x - 1)^4), {x, 0, 50}], x] (* Wesley Ivan Hurt, Nov 12 2022 *) -
PARI
a(n)=if(n==3, 0, -1 - floor((n-1)/2) + n * sum(k=1,floor(n/3), floor((n-3*k+2)/2))) \\ Winston de Greef, Jan 28 2024
Formula
For n >= 4, a(n) = -1 - floor((n-1)/2) + n * Sum_{k=1..floor(n/3)} floor((n-3*k+2)/2).
G.f.: -x^4 * (x^9-x^8-3*x^7+5*x^5+6*x^4-6*x^3-11*x^2-7*x-2) / ((x+1)^2 *(x^2+x+1)^2 *(x-1)^4). - Alois P. Heinz, Dec 13 2021
a(n) = 2*a(n-2)+2*a(n-3)-a(n-4)-4*a(n-5)-a(n-6)+2*a(n-7)+2*a(n-8)-a(n-10). - Wesley Ivan Hurt, Dec 17 2021