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%I #11 Dec 23 2024 09:53:47
%S 1,3,6,11,14,21,24,31,36,39,46,53,56,63,70,75,78,85,92,99,102,109,116,
%T 123,128,131,138,145,152,159,162,169,176,183,190,195,198,205,212,219,
%U 226,233,236,243,250,257,264,271,276,279,286,293,300,307,314,321,324,331,338
%N 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,... .
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F 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.
%e ---------------------------------------------------------------------------
%e partitions of k into two parts
%e ---------------------------------------------------------------------------
%e k 2 3 4 5 6 7 8 9
%e [1,7] [1,8]
%e [1,5] [1,6] [2,6] [2,7]
%e [1,3] [1,4] [2,4] [2,5] [3,5] [3,6]
%e [1,1] [1,2] [2,2] [2,3] [3,3] [3,4] [4,4] [4,5]
%e ----------------------------------------------------------------------------
%e n circle centers number of regions formed
%e ----------------------------------------------------------------------------
%e 1 (1,1) 1
%e 2 (1,1), (1,2) 3
%e 3 (1,1), (1,2), (1,3) 6
%e 4 (1,1), (1,2), (1,3), (2,2) 11
%e 5 (1,1), (1,2), (1,3), (2,2), (1,4) 14
%e 6 (1,1), (1,2), (1,3), (2,2), (1,4), (2,3) 21
%e ...
%t 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}]]
%Y Cf. A339399.
%K nonn
%O 1,2
%A _Wesley Ivan Hurt_, Mar 30 2021