A337774 Number of partitions of 2*n into two parts, (s,t), such that at least one of s+-1 or t+-1 is a square > 0.

0, 1, 2, 2, 3, 4, 5, 5, 4, 5, 6, 7, 7, 8, 9, 7, 7, 8, 9, 9, 10, 11, 11, 11, 9, 10, 12, 12, 11, 12, 13, 13, 13, 13, 14, 13, 13, 14, 14, 13, 14, 16, 17, 16, 15, 16, 17, 17, 15, 15, 16, 16, 16, 18, 19, 19, 18, 17, 18, 19, 19, 20, 21, 18, 17, 19, 20, 19, 20, 21, 21, 20, 19, 20, 22

Offset

1,3

Links

Table of n, a(n) for n=1..75.

Formula

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

Example

a(6) = 4; There are 6 partitions of 2*6 = 12 into two parts, (11,1), (10,2), (9,3), (8,4), (7,5) and (6,6). Since 10-1 = 9 (square), 3+1 = 4 (square), 8+1 = 9 (square), and 5-1 = 4 (square), then a(6) = 4.

Mathematica

Table[Sum[Sign[(Floor[Sqrt[i - 1]] - Floor[Sqrt[i - 2]]) + (Floor[Sqrt[2 n - i - 1]] - Floor[Sqrt[2 n - i - 2]]) + (Floor[Sqrt[i + 1]] - Floor[Sqrt[i]]) + (Floor[Sqrt[2 n - i + 1]] - Floor[Sqrt[2 n - i]])], {i, 2, n}], {n, 100}]

Crossrefs

Cf. A010052.

Sequence in context: A320159 A342879 A342880 * A371263 A069928 A135585

Adjacent sequences: A337771 A337772 A337773 * A337775 A337776 A337777

Keyword

nonn

Author

_Wesley Ivan Hurt_, Sep 19 2020

Status

approved