A024813 Positive integers m for which f(m-1) < f(m) < f(m+1), where f(m) = floor(cot(Pi/(2m))).

4, 7, 12, 15, 18, 23, 26, 29, 34, 37, 40, 45, 48, 51, 54, 59, 62, 65, 70, 73, 76, 81, 84, 87, 92, 95, 98, 103, 106, 109, 114, 117, 120, 125, 128, 131, 136, 139, 142, 147, 150, 153, 158, 161, 164, 169, 172, 175, 180, 183, 186, 191, 194, 197, 202, 205, 208, 213, 216, 219, 224, 227

Offset

1,1

Comments

Conjecture (verified for m < 10^6 by M. F. Hasler): A024813(n) = 2*A024812(n) - n + 1, n=1,2,.... - L. Edson Jeffery, Mar 21 2013

The above conjecture follows from the Laurent series for cot(x) = 1/x - x/3 + O(x^3) and the conjecture n/a(n) ~ 4/Pi-1. - M. F. Hasler, Mar 25 2013

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Robert G. Wilson v, Graph of f(n), see Mathematica program

Formula

n/a(n) ~ 4/Pi - 1 (as n -> oo), or a(n) ~ 3.65979 n. (Conjectured.) - M. F. Hasler, Mar 25 2013

Alternate formula: cot(Pi/(2m)) = tan((Pi/2)*(1 - 1/m)).

Conjecture: a(n) = a(n-1) + a(n-3) - a(n-4); g.f.: x*(2*x^15-2*x^14-x^3+5*x^2+3*x+4) / ((x-1)^2*(x^2+x+1)). - Colin Barker, Jan 03 2014

Mathematica

f[n_] := Floor[Tan[Pi (1 - 1/n)/2]]; Select[Range[2, 100], f[# - 1] < f[#] < f[# + 1] &] (* Robert G. Wilson v, Mar 19 2013 *)
Flatten[Position[Partition[Floor[Cot[Pi/(2Range[250])]], 3, 1], _?(Min[ Differences[ #]]>0&), {1}, Heads->False]]+1 (* Harvey P. Dale, Feb 04 2016 *)

Prog

(PARI) {my(f(m)=floor(cotan(Pi/2/m))); for(m=2, 99, f(m-1)<f(m) & f(m)<f(m+1) & print1(m", "))} \\ See comment in A024812; M. F. Hasler, Mar 20 2013

Crossrefs

A024812 yields the corresponding values of f.

Sequence in context: A268816 A161843 A085048 * A276880 A276875 A310771

Adjacent sequences: A024810 A024811 A024812 * A024814 A024815 A024816

Keyword

nonn

Author

Clark Kimberling

Extensions

Definition corrected by M. F. Hasler, following posts to the SeqFan list from Harvey P. Dale and Don Reble, Mar 19 2013

Status

approved