A393095 Group the partitions of n into 3 parts into sets whose smallest parts are equal. a(n) is the sum of the minimum values of all the largest parts in each set.

0, 0, 0, 1, 2, 2, 5, 6, 7, 11, 13, 14, 20, 22, 24, 31, 34, 36, 45, 48, 51, 61, 65, 68, 80, 84, 88, 101, 106, 110, 125, 130, 135, 151, 157, 162, 180, 186, 192, 211, 218, 224, 245, 252, 259, 281, 289, 296, 320, 328, 336, 361, 370, 378, 405, 414, 423, 451, 461, 470, 500, 510, 520, 551, 562, 572, 605, 616, 627, 661, 673, 684, 720, 732, 744, 781

Offset

0,5

Links

Table of n, a(n) for n=0..75.

Index entries for sequences related to partitions

Index entries for linear recurrences with constant coefficients, signature (0,1,2,0,-2,-1,0,1).

Formula

G.f.: x^3*(1+2*x+x^2+x^3)/((1-x)^3*(1+x)*(1+x+x^2)^2).

a(n) = a(n-2) + 2*a(n-3) - 2*a(n-5) - a(n-6) + a(n-8).

a(n) = Sum_{j=1..floor(n/3)} ceiling((n-j)/2).

a(n) = floor(n/3)*(2*n-1-floor(n/3))/4-floor((n-2)/3)/2+floor((n-2)/2)/2.

a(n) = A211546(n) - A393093(n).

Example

a(10) = 13; The partitions of 10 into 3 parts grouped by smallest part: {(8,1,1),(7,2,1),(6,3,1),(5,4,1)}, {(6,2,2),(5,3,2),(4,4,2)}, {(4,3,3)}. The sum of the minimum values of all the largest parts in each set is 5+4+4 = 13.

Mathematica

CoefficientList[Series[x^3*(1 + 2*x + x^2 + x^3)/((1 - x)^3*(1 + x)*(1 + x + x^2)^2), {x, 0, 80}], x]
LinearRecurrence[{0, 1, 2, 0, -2, -1, 0, 1}, {0, 0, 0, 1, 2, 2, 5, 6}, 80]
Table[Floor[n/3] (2 n - 1 - Floor[n/3])/4 - Floor[(n - 2)/3]/2 + Floor[(n - 2)/2]/2, {n, 0, 80}]

Crossrefs

Cf. A069905, A211546, A393093 (sum of maximums of middle parts).

Sequence in context: A054255 A063177 A249521 * A355345 A034803 A205885

Adjacent sequences: A393092 A393093 A393094 * A393096 A393097 A393098

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Feb 01 2026

Status

approved