A339183 Number of partitions of n into two parts such that the smaller part is a nonzero square.

0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7

Offset

0,9

Comments

Every nonnegative integer k appears in blocks of 4k+2 k's.

Links

Table of n, a(n) for n=0..98.

Index entries for sequences related to partitions

Formula

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

a(n) = A339186(n) - A339184(n).

a(n) = floor(sqrt(n/2)), for n >= 0. [See the comment, and {4*k+2 = A016825 (k)} is the first difference sequence of the sequence b(k) = 2*k^2 = A001105(k) for which sqrt(b(k)/2) becomes an integer.] - Wolfdieter Lang, Dec 07 2020

Example

a(7) = 1; The partitions of 7 into two parts are (6,1), (5,2) and (4,3). Since 1 is the only nonzero square appearing among the smallest parts, a(7) = 1.
a(8) = 2; The partitions of 8 into two parts are (7,1), (6,2), (5,3) and  (4,4). Since 1 and 4 are the only nonzero squares appearing among the smallest parts, a(8) = 2.

Mathematica

Table[Sum[Floor[Sqrt[i]] - Floor[Sqrt[i - 1]] , {i, Floor[n/2]}], {n, 0, 120}]

Crossrefs

Cf. A010052, A339184 (larger part is a nonzero square), A339186 (total number of nonzero squares).

Cf. A001105, A016825.

Sequence in context: A185711 A185712 A328499 * A135665 A135662 A111853

Adjacent sequences: A339180 A339181 A339182 * A339184 A339185 A339186

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Nov 26 2020

Status

approved