A339184 Number of partitions of n into two parts such that the larger part is a nonzero square.

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, 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, 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

Offset

0,18

Links

Table of n, a(n) for n=0..92.

Index entries for sequences related to partitions

Formula

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

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

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

Example

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.
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.

Mathematica

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

Crossrefs

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

Sequence in context: A347735 A204014 A378397 * A156839 A172299 A025453

Adjacent sequences: A339181 A339182 A339183 * A339185 A339186 A339187

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Nov 26 2020

Status

approved