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

3,2

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).

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A069905, A350042.

Sequence in context: A070898 A132746 A252475 * A167543 A332495 A184976

Adjacent sequences: A350040 A350041 A350042 * A350044 A350045 A350046

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Dec 10 2021

Status

approved