%I #6 Dec 07 2020 01:37:13

%S 0,0,1,0,0,1,1,1,1,0,1,1,1,1,1,1,1,2,2,1,1,1,1,1,1,1,2,2,2,2,2,2,2,1,

%T 1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,

%U 3,3,3,3,3,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3

%N Number of partitions of n into two parts such that the larger part is a nonzero square.

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

%F a(n) = Sum_{i=1..floor(n/2)} c(n-i), where c is the square characteristic (A010052).

%F a(n) = Sum_{i=floor((n-1)/2)..n-2} c(i+1), where c is the square characteristic (A010052).

%F a(n) = A339186(n) - A339183(n).

%e a(8) = 1; The partitions of 8 into 2 parts are (7,1), (6,2), (5,3) and (4,4). Since 4 is the only nonzero square appearing as a largest part, a(8) = 1.

%e a(9) = 0; The partitions of 9 into 2 parts are (8,1), (7,2), (6,3) and (5,4). Since there are no nonzero squares among the largest parts, a(9) = 0.

%t Table[Sum[Floor[Sqrt[n - i]] - Floor[Sqrt[n - i - 1]] , {i, Floor[n/2]}], {n, 0, 100}]

%Y Cf. A010052, A339183 (smaller part is a nonzero square), A339186 (total nonzero squares).

%K nonn,easy

%O 0,18

%A _Wesley Ivan Hurt_, Nov 26 2020