A392567 Sum of the smallest parts in the partitions of n into 3 parts whose middle part is even.

0, 0, 0, 0, 0, 1, 3, 3, 3, 4, 6, 9, 13, 14, 16, 19, 23, 29, 37, 40, 44, 50, 58, 68, 80, 86, 94, 104, 116, 131, 149, 159, 171, 186, 204, 225, 249, 264, 282, 303, 327, 355, 387, 408, 432, 460, 492, 528, 568, 596, 628, 664, 704, 749, 799, 835, 875, 920, 970, 1025, 1085, 1130, 1180, 1235, 1295, 1361, 1433, 1488, 1548, 1614, 1686, 1764, 1848

Offset

0,7

Links

Table of n, a(n) for n=0..72.

Index entries for sequences related to partitions

Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1,0,2,-4,4,-4,2,0,-1,2,-2,2,-1).

Formula

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

a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) + 2*a(n-6) - 4*a(n-7) + 4*a(n-8) - 4*a(n-9) + 2*a(n-10) - a(n-12) + 2*a(n-13) - 2*a(n-14) + 2*a(n-15) - a(n-16).

a(n) + A309690(n) + A392475(n) = n*A309689(n). - Wesley Ivan Hurt, Feb 17 2026

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

Example

a(11) = 9; There are 5 partitions of 11 into 3 parts whose middle part is even: (8,2,1), (6,4,1), (7,2,2), (5,4,2) and (4,4,3). The sum of the smallest parts of these partitions is 1+1+2+2+3 = 9.

Mathematica

Table[Sum[Sum[j*Mod[k + 1, 2], {k, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 100}]
LinearRecurrence[{2, -2, 2, -1, 0, 2, -4, 4, -4, 2, 0, -1, 2, -2, 2, -1}, {0, 0, 0, 0, 0, 1, 3, 3, 3, 4, 6, 9, 13, 14, 16, 19}, 100]

Prog

(PARI) a(n)=(2*n^3+9*n^2+12*[6, 1, -2, -3, -2, 1, 6, 1, -2, -3, -2, 1][n%12+1]*n+[0, -23, -4, -27, -176, -103, 108, 85, -112, -135, -68, 5][n%12+1])/432 \\ Charles R Greathouse IV, May 31 2026

Crossrefs

Cf. A309689, A309690, A392475.

Sequence in context: A238505 A259897 A091849 * A035567 A219328 A108688

Adjacent sequences: A392564 A392565 A392566 * A392568 A392569 A392570

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Feb 13 2026

Status

approved