

A341319


Minimum number of cells needed in an n X n grid to win a majority of districts when all districts are required to be the same size.


4



1, 3, 4, 9, 9, 14, 16, 25, 25, 33, 36, 45, 49, 60, 64, 81, 81, 95, 100, 117, 121, 138, 144, 165, 169, 189, 196, 225, 225, 247, 256, 289, 289, 315, 324, 350, 361, 390, 400, 429, 441, 473, 484, 529, 529, 564, 576, 625, 625, 663, 676, 729, 729, 770, 784, 825, 841, 885, 900, 943
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OFFSET

1,2


COMMENTS

Consider an n X n grid in which each cell represents a vote for a political party "A" or "B". The grid is divided into equalsized districts and the winner of each district is decided by a majority of "A"s or "B"s. This sequence is the minimum number of "A"s needed to win more districts than "B" if n = 1, 2, 3, ... A tie within a district is not accepted. For example, if n=5, and the district size is also 5, party "A" needs 3 cells in 3 districts (total = 3*3=9) to win 3 districts to 2.
This is related to the gerrymandering question.  N. J. A. Sloane, Feb 27 2021


LINKS



FORMULA

a(n) is the minimum value of (floor(d/2)+1)*(floor(n^2/(2*d))+1) over all divisors d of n^2.


EXAMPLE

For a(3), divisors of 3^2 are 1, 3, 9:
d=1: (floor(1/2)+1)*(floor(3^2/(2*1))+1) = 1*5 = 5
d=3: (floor(3/2)+1)*(floor(3^2/(2*3))+1) = 2*2 = 4
d=9: (floor(9/2)+1)*(floor(3^2/(2*9))+1) = 5*1 = 5
Party A only needs 4 cells out of 9 to win a majority of districts.
For a(6), divisors of 6^2 are 1, 2, 3, 4, 6, 9, 12, 18, 36:
By symmetry we can ignore d = 9, 12, 18 and 36;
d=1: (floor(1/2)+1)*(floor(6^2/(2*1))+1) = 1*19 = 19
d=2: (floor(2/2)+1)*(floor(6^2/(2*2))+1) = 2*10 = 20
d=3: (floor(3/2)+1)*(floor(6^2/(2*3))+1) = 2*7 = 14
d=4: (floor(4/2)+1)*(floor(6^2/(2*4))+1) = 3*5 = 15
d=6: (floor(6/2)+1)*(floor(6^2/(2*6))+1) = 4*4 = 16
Party A only needs 14 cells out of 36 to win a majority of districts.


MAPLE

a:= n>min(map(d>(iquo(d, 2)+1)*(iquo(n^2, 2*d)+1), numtheory[divisors](n^2))):


MATHEMATICA

a[n_] := Table[(Floor[d/2]+1)*(Floor[n^2/(2d)]+1), {d, Divisors[n^2]}] // Min;


PROG

(PARI) a(n)={vecmin([(floor(d/2) + 1)*(floor(n^2/(2*d)) + 1)  d<divisors(n^2)])} \\ Andrew Howroyd, Feb 09 2021
(Python)
from sympy import divisors
def A341319(n): return min((d//2+1)*(e//2+1) for d, e in ((v, n**2//v) for v in divisors(n**2) if v <= n)) # Chai Wah Wu, Mar 05 2021


CROSSREFS

Cf. A341578 (ties are allowed), A002265(n+6) (US Electoral College system: unequal sizes of districts, winner in each district takes all its votes), A290323 (multiple levels).
See A341721 for a case where the number of voters need not be a square.


KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



