A393402 - OEIS

%I #7 May 31 2026 21:53:20

%S 0,0,0,1,1,1,1,2,4,7,7,8,10,13,17,23,25,28,32,38,46,56,60,66,74,84,96,

%T 111,119,129,141,156,174,195,207,222,240,261,285,313,331,352,376,404,

%U 436,472,496,524,556,592,632,677,709,745,785,830,880,935,975,1020,1070,1125,1185,1251,1301,1356,1416,1482,1554,1632,1692

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

%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 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) + A309688(n) + A393412(n) = n*A309687(n). - _Wesley Ivan Hurt_, Feb 17 2026

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

%e a(10) = 7; There are 4 partitions of 10 into 3 parts whose middle part is odd: (8,1,1), (6,3,1), (5,3,2) and (4,3,3).  The sum of the smallest parts of these partitions is 1+1+2+3 = 7.

%t Table[Sum[Sum[j*Mod[k, 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, 1, 1, 1, 1, 2, 4, 7, 7, 8, 10, 13, 17, 23}, 100]

%o (PARI) a(n)=(2*n^3+9*n^2+12*[-3,-2,1,6,1,-2,-3,-2,1,6,1,-2][n%12+1]*n+[0,13,-76,81,112,77,-108,-95,32,189,4,-31][n%12+1])/432 \\ _Charles R Greathouse IV_, May 31 2026

%Y Cf. A309687, A309688, A393412.

%K nonn,easy

%O 0,8

%A _Wesley Ivan Hurt_, Feb 13 2026