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A380377 Minimum number of total votes needed for one party to win if there are n voters divided into balanced districts, i.e., the numbers of voters in two districts may differ by at most 1.

%I A380377 #7 Jan 26 2025 21:01:38
%S A380377 1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,6,7,7,8,8,8,8,8,9,9,9,10,10,10,10,11,
%T A380377 12,12,12,12,12,12,13,14,14,14,14,14,14,15,15,15,15,16,16,16,17,18,18,
%U A380377 18,18,18,18,18,19,20,20,20,20,20,20,21,21,21,21,22
%N A380377 Minimum number of total votes needed for one party to win if there are n voters divided into balanced districts, i.e., the numbers of voters in two districts may differ by at most 1.
%C A380377 The rules are the same as in A341721 (except that the number of voters in two districts may differ by 1 here): The winner must have a strict majority of the votes in a strictly larger number of districts than the other party has.
%C A380377 Empirically, it seems that the limit of (a(n)-n/4)/sqrt(n) exists with an approximate value of 0.3538.
%H A380377 Pontus von Brömssen, <a href="/A380377/b380377.txt">Table of n, a(n) for n = 1..10000</a>
%H A380377 Pontus von Brömssen, <a href="/A380378/a380378.png">Illustration for a(100000)=25116</a>.
%H A380377 Wikipedia, <a href="https://en.wikipedia.org/wiki/Gerrymandering">Gerrymandering</a>.
%F A380377 a(n) <= A341721(n).
%F A380377 a(n) = a(n-1)+1 if n is in A380379, otherwise a(n) = a(n-1).
%F A380377 a(n) = A380378(n,A380381(n)) = A380378(n,A380382(n)).
%e A380377 For n = 9, a(9) = 4 votes are required to win. There can be either 3 districts 3+3+3 with 2 supporters in 2 of them, 6 districts 1+1+1+2+2+2 with 3 supporters in the single-voter districts and 1 in a 2-voter district, or 7 districts 1+1+1+1+1+2+2 with supporters in 4 of the single-voter districts.
%e A380377 For n = 17, a(17) = 6 votes are required to win. This can only be achieved with 5 districts 3+3+3+4+4 with 2 supporters in each of the 3 smaller districts.
%Y A380377 Row minima of A380378.
%Y A380377 Cf. A341721, A380379, A380380, A380381, A380382, A380383.
%K A380377 nonn
%O A380377 1,2
%A A380377 _Pontus von Brömssen_, Jan 24 2025