A384717 Sum of floored squared chord lengths from -1 to the 2n-th roots of unity (upper semicircle, endpoints excluded).

0, 2, 4, 5, 6, 9, 9, 11, 13, 14, 15, 18, 18, 20, 22, 23, 24, 27, 27, 29, 31, 32, 33, 36, 36, 38, 40, 41, 42, 45, 45, 47, 49, 50, 51, 54, 54, 56, 58, 59, 60, 63, 63, 65, 67, 68, 69, 72, 72, 74, 76, 77, 78, 81, 81, 83, 85, 86, 87, 90

Offset: 1

Author

Joost de Winter, Aug 19 2025

Keywords

• nonn
• easy
• new

Comments

Let z_k = exp(i*Pi*k/n) for k = 1..n-1 (upper 2n-th roots, excluding endpoints +-1). By Euler's formula, |1 + z_k|^2 = 2 + 2*cos(Pi*k/n). The sequence a(n) is Sum_{k=1..n-1} floor(2 + 2*cos(Pi*k/n)), i.e., the sum of floored squared chord lengths from -1 to those points.

Distribution of the summands: floor(2 + 2*cos(Pi*k/n)) takes values 3, 2, 1, 0 exactly floor(n/3), floor(n/2) - floor(n/3), floor(2*n/3) - floor(n/2), (n-1) - floor(2*n/3) times, respectively.

If one includes the endpoints (k = 0 and k = n), the resulting total is a(n) + 4.

Examples

			n = 10: the list floor(2 + 2*cos(Pi*k/10)) for k = 1..9 is 3, 3, 2, 2, 2, 1, 1, 0, 0; sum = 14.
Check with the closed form: floor(10/2) + floor(10/3) + floor(20/3) = 5 + 3 + 6 = 14.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,-1).

Crossrefs

Cf. A010761, A004523.

Programs

MATLAB

function an = euler_chord_term(n)
    an = floor(n/2) + floor(n/3) + floor(2*n/3);
end

Mathematica

A384717[n_] := Quotient[n, 2] + Quotient[n, 3] + Quotient[2*n, 3];
Array[A384717, 100] (* Paolo Xausa, Aug 28 2025 *)

PARI

a(n) = n\2 + n\3 + 2*n\3; \\ Amiram Eldar, Aug 29 2025

Formula

a(n) = floor(n/2) + floor(n/3) + floor(2*n/3).