A342954 Number of interior regions formed by drawing n unit circles whose centers (s,t) are the partitions of k into 2 parts, where 0 < s <= t and s + t = k, ordered by increasing values of s and where k = 2,3,... .

1, 3, 6, 11, 14, 21, 24, 31, 36, 39, 46, 53, 56, 63, 70, 75, 78, 85, 92, 99, 102, 109, 116, 123, 128, 131, 138, 145, 152, 159, 162, 169, 176, 183, 190, 195, 198, 205, 212, 219, 226, 233, 236, 243, 250, 257, 264, 271, 276, 279, 286, 293, 300, 307, 314, 321, 324, 331, 338

Offset

1,2

Links

Table of n, a(n) for n=1..59.

Index entries for sequences related to partitions

Formula

a(1) = 1; a(n) = Sum_{i=1..n-1} (7 - 4*(floor(sqrt(i)) - floor(sqrt(i - 1)) + floor(sqrt(i + 1) + 1/2) - floor(sqrt(i) + 1/2)) - 2*(floor(sqrt(i + 1)) - floor(sqrt(i)))) for n > 1.

Example

---------------------------------------------------------------------------
                       partitions of k into two parts
---------------------------------------------------------------------------
   k       2       3       4       5       6       7       8       9
                                                         [1,7]   [1,8]
                                         [1,5]   [1,6]   [2,6]   [2,7]
                         [1,3]   [1,4]   [2,4]   [2,5]   [3,5]   [3,6]
         [1,1]   [1,2]   [2,2]   [2,3]   [3,3]   [3,4]   [4,4]   [4,5]
----------------------------------------------------------------------------
   n               circle centers            number of regions formed
----------------------------------------------------------------------------
   1       (1,1)                                         1
   2       (1,1), (1,2)                                  3
   3       (1,1), (1,2), (1,3)                           6
   4       (1,1), (1,2), (1,3), (2,2)                   11
   5       (1,1), (1,2), (1,3), (2,2), (1,4)            14
   6       (1,1), (1,2), (1,3), (2,2), (1,4), (2,3)     21
  ...

Mathematica

Join[{1}, Table[Sum[7 - 4 (Floor[Sqrt[i]] - Floor[Sqrt[i - 1]] + Floor[Sqrt[i + 1] + 1/2] - Floor[Sqrt[i] + 1/2]) - 2 (Floor[Sqrt[i + 1]] - Floor[Sqrt[i]]), {i, n - 1}], {n, 2, 80}]]

Crossrefs

Cf. A339399.

Sequence in context: A342173 A282277 A122599 * A015823 A049620 A350438

Adjacent sequences: A342951 A342952 A342953 * A342955 A342956 A342957

Keyword

nonn,changed

Author

Wesley Ivan Hurt, Mar 30 2021

Status

approved