A392567 - OEIS

%I #15 May 31 2026 21:53:08

%S 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,

%T 104,116,131,149,159,171,186,204,225,249,264,282,303,327,355,387,408,

%U 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

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

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%H <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-1,0,2,-4,4,-4,2,0,-1,2,-2,2,-1).

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

%F 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).

%F a(n) + A309690(n) + A392475(n) = n*A309689(n). - _Wesley Ivan Hurt_, Feb 17 2026

%F a(n) ~ n^3/216. - _Charles R Greathouse IV_, May 31 2026

%e 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.

%t Table[Sum[Sum[j*Mod[k + 1, 2], {k, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 100}]

%t 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]

%o (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

%Y Cf. A309689, A309690, A392475.

%K nonn,easy

%O 0,7

%A _Wesley Ivan Hurt_, Feb 13 2026