A381194 - OEIS

%I #20 Apr 13 2025 01:47:26

%S 1,3,3,9,5,17,7,33,15,49,11,113,13,153,57,321,17,617,19,1153,165,2089,

%T 23,4577,85,8241,555,16737,29,34049,31,66177,2109,131137,377,267521,

%U 37,524361,8265,1051393,41,2114081,43,4198561,33945,8388697,47,16851905,427,33556689

%N Number of equal-length matchings of 2n uniformly spaced points on a circle.

%C a(n) is the number of ways to draw n segments of equal length so that each vertex of a regular 2n-gon is an endpoint of one of the segments. It is not difficult to prove that the Maple program given below computes the correct value. Note that a(n) = n when n is an odd prime.

%C Finding the value when n = 12 was a problem in the 2025 American Invitational Mathematics Examination (AIME), a part of the American Mathematics Competitions run by the Mathematical Association of America. A solution is presented in the Art of Problem Solving website.

%H Yifan Xie, <a href="/A381194/b381194.txt">Table of n, a(n) for n = 1..5000</a>

%H Art of Problem Solving, <a href="https://artofproblemsolving.com/wiki/index.php/2025_AIME_II_Problems/Problem_11">2025 AIME II Problems/Problem 11</a>.

%F a(n) = 1 + Sum_{i=1..n-1} [2*n/GCD(2*n,i) mod 2 == 0] * 2^GCD(2*n,i), where [ ] denotes the Iverson bracket.

%p a := 1;

%p for i from 1 to n - 1 do

%p     if (2*n/gcd(2*n, i)) mod 2 = 0 then a := a + 2^gcd(2*n, i); end if;

%p end do;

%t a[n_]:=1+Sum[Boole[Mod[2n/GCD[2n,i],2]==0]2^GCD[2n,i],{i,n-1}]; Array[a,50] (* _Stefano Spezia_, Feb 24 2025 *)

%o (PARI) a(n)={my(s=1); for(i=1, n-1, my(d=gcd(2*n, i)); if((2*n/d)%2 == 0, s += 2^d)); s} \\ _Yifan Xie_, Apr 13 2025

%K nonn

%O 1,2

%A _Jerrold Grossman_, Feb 16 2025