A392474 Sum of the largest parts in the partitions of n into 3 parts whose smallest part is even.

0, 0, 0, 0, 0, 0, 2, 3, 7, 9, 15, 18, 30, 35, 51, 58, 78, 87, 117, 129, 165, 180, 222, 240, 296, 318, 382, 408, 480, 510, 600, 635, 735, 775, 885, 930, 1062, 1113, 1257, 1314, 1470, 1533, 1715, 1785, 1981, 2058, 2268, 2352, 2592, 2684, 2940, 3040, 3312, 3420, 3726, 3843, 4167, 4293, 4635, 4770, 5150, 5295, 5695, 5850, 6270

Offset

0,7

Links

Table of n, a(n) for n=0..64.

Index entries for sequences related to partitions

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

Formula

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

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

a(n) + A309686(n) + A393408(n) = n*A309685(n). - Wesley Ivan Hurt, Feb 17 2026

a(n) ~ 11*n^3/432. - Charles R Greathouse IV, May 31 2026

Example

a(11) = 18; There are 3 partitions of 11 into 3 parts whose smallest part is even: (7,2,2), (6,3,2), and (5,4,2). The sum of the largest parts of these partitions is 7+6+5 = 18.

Mathematica

Table[Sum[Sum[(n - j - k)*Mod[j + 1, 2], {k, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 100}]
LinearRecurrence[{1, 1, -1, 0, 0, 2, -2, -2, 2, 0, 0, -1, 1, 1, -1}, {0, 0, 0, 0, 0, 0, 2, 3, 7, 9, 15, 18, 30, 35, 51}, 100]

Prog

(PARI) a(n)=my(q=n\6); 11*q^3/2+q*[-7*q, -4*q-1, 4*q-1, 7*q, 15*q+4, 18*q+7][n%6+1]/2 \\ Charles R Greathouse IV, May 31 2026

Crossrefs

Cf. A309685, A309686, A393408.

Sequence in context: A294122 A211539 A109660 * A236544 A343198 A075855

Adjacent sequences: A392471 A392472 A392473 * A392475 A392476 A392477

Keyword

nonn,easy,changed

Author

Wesley Ivan Hurt, Feb 13 2026

Status

approved