

A339183


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


2



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
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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



