A393093 Group the partitions of n into 3 parts into sets whose smallest parts are equal. a(n) is the sum of the maximum values of all the middle parts in each set.

0, 0, 0, 1, 1, 2, 4, 5, 6, 10, 11, 13, 18, 20, 22, 29, 31, 34, 42, 45, 48, 58, 61, 65, 76, 80, 84, 97, 101, 106, 120, 125, 130, 146, 151, 157, 174, 180, 186, 205, 211, 218, 238, 245, 252, 274, 281, 289, 312, 320, 328, 353, 361, 370, 396, 405, 414, 442, 451, 461, 490, 500, 510, 541, 551, 562, 594, 605, 616, 650, 661, 673, 708, 720

Offset

0,6

Links

Table of n, a(n) for n=0..73.

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 + x + x^2 + x^3 + x^4)/((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)} floor((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) = A069905(n) + A211534(n).

a(n) = A211546(n) - A393095(n).

Example

a(10) = 11; 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 maximum values of all the middle parts in each set is 4+4+3 = 11.

Mathematica

CoefficientList[Series[x^3*(1 + x + x^2 + x^3 + x^4)/((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, 1, 2, 4, 5}, 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, A211534, A211546, A393095 (sum of minimums of largest parts).

Sequence in context: A117890 A108853 A257085 * A334736 A265349 A047433

Adjacent sequences: A393090 A393091 A393092 * A393094 A393095 A393096

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Feb 01 2026

Status

approved