A393403 - OEIS

%I #7 May 31 2026 21:53:46

%S 0,0,0,1,0,1,1,4,3,7,4,10,9,17,14,25,19,32,29,46,40,60,50,74,68,96,86,

%T 119,104,141,131,174,159,207,186,240,225,285,264,331,303,376,355,436,

%U 408,496,460,556,528,632,596,709,664,785,749,880,835,975,920,1070,1025,1185,1130,1301,1235,1416,1361,1554,1488,1692,1614

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

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

%H <a href="/index/Rec#order_17">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1,2,-2,0,0,-2,2,-1,1,0,0,1,-1).

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

%F 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).

%F a(n) + A393406(n) + A309692(n) = n*A026923(n). - _Wesley Ivan Hurt_, Feb 17 2026

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

%e a(11) = 10; There are 6 partitions of 11 into 3 parts whose largest part is odd: (9,1,1), (7,3,1), (5,5,1), (7,2,2), (5,4,2) and (5,3,3).  The sum of the smallest parts of these partitions is 1+1+1+2+2+3 = 10.

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

%t LinearRecurrence[{1, 0, 0, 1, -1, 2, -2, 0, 0, -2, 2, -1, 1, 0, 0, 1, -1}, {0, 0, 0, 1, 0, 1, 1, 4, 3, 7, 4, 10, 9, 17, 14, 25, 19}, 100]

%o (PARI) a(n)=(2*n^3+(3+n%2*12)*n^2+12*[0,3,0,3,-4,-1,0,3,0,3,-4,-1][n%12+1]*n+[0,-53,-28,135,16,-133,-108,55,80,27,-92,-25][n%12+1])/432 \\ _Charles R Greathouse IV_, May 31 2026

%Y Cf. A026923, A309692, A393406.

%K nonn,easy

%O 0,8

%A _Wesley Ivan Hurt_, Feb 13 2026