A393404 Sum of the smallest parts in the partitions of n into 3 parts whose largest part is even.

0, 0, 0, 0, 1, 1, 3, 1, 4, 4, 9, 7, 14, 10, 19, 17, 29, 25, 40, 32, 50, 46, 68, 60, 86, 74, 104, 96, 131, 119, 159, 141, 186, 174, 225, 207, 264, 240, 303, 285, 355, 331, 408, 376, 460, 436, 528, 496, 596, 556, 664, 632, 749, 709, 835, 785, 920, 880, 1025, 975, 1130, 1070, 1235, 1185, 1361, 1301, 1488, 1416, 1614, 1554, 1764

Offset

0,7

Links

Table of n, a(n) for n=0..70.

Index entries for sequences related to partitions

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

Formula

a(n) = Sum_{j=1..floor(n/3)} Sum_{k=j..floor((n-j)/2)} j * ((n-j-k+1) mod 2).

a(n) = a(n-1) + a(n-4) - a(n-5) + 2*a(n-6) - 2*a(n-7) - 2*a(n-10) + 2*a(n-11) - a(n-12) + a(n-13) + a(n-16) - a(n-17).

a(n) + A393409(n) + A309694(n) = n*A026927(n). - Wesley Ivan Hurt, Feb 17 2026

a(n) ~ n^3/216. - Charles R Greathouse IV, May 31 2026

Example

a(10) = 9; There are 5 partitions of 10 into 3 parts whose largest part is even: (8,1,1), (6,3,1), (6,2,2), (4,4,2) and (4,3,3). The sum of the smallest parts of these partitions is 1+1+2+2+3 = 9.

Mathematica

Table[Sum[Sum[j*Mod[n - j - k + 1, 2], {k, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 100}]
LinearRecurrence[{1, 0, 0, 1, -1, 2, -2, 0, 0, -2, 2, -1, 1, 0, 0, 1, -1}, {0, 0, 0, 0, 1, 1, 3, 1, 4, 4, 9, 7, 14, 10, 19, 17, 29}, 100]

Prog

(PARI) a(n)=(2*n^3+(15-n%2*12)*n^2+12*[3, -4, -1, 0, 3, 0, 3, -4, -1, 0, 3, 0][n%12+1]*n+[0, 43, -52, -81, -80, 107, 108, -65, -160, 27, 28, -1][n%12+1])/432 \\ Charles R Greathouse IV, May 31 2026

Crossrefs

Cf. A026927, A309694, A393409.

Sequence in context: A363569 A200027 A370807 * A298890 A016473 A343516

Adjacent sequences: A393401 A393402 A393403 * A393405 A393406 A393407

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Feb 13 2026

Status

approved