A237128 Angles n expressed in degrees such that 2*cos(n) = phi where phi is the golden ratio (A001622).

36, 324, 396, 684, 756, 1044, 1116, 1404, 1476, 1764, 1836, 2124, 2196, 2484, 2556, 2844, 2916, 3204, 3276, 3564, 3636, 3924, 3996, 4284, 4356, 4644, 4716, 5004, 5076, 5364, 5436, 5724, 5796, 6084, 6156, 6444, 6516, 6804, 6876, 7164, 7236, 7524, 7596, 7884

Offset: 1

Author

Michel Lagneau, Feb 04 2014

Keywords

• nonn
• easy

Comments

a(n) == 36, 324 mod 360 and a(n)/36 is congruent to {1,9} mod 10 (A090771).

See A019863 = half of the golden ratio (A001622) => a(1) = 90 - 54 degrees and a(2) = 360 - a(1) = 324 degrees.

The squares in the sequence are 36, 324, 1764, 2916, 4356, 6084, 10404, 12996, 15876, 19044, 26244, 30276, 34596, 39204, 49284, 54756, 60516, 66564, 79524,... with the following properties:

If a(n) == 36 mod 360 is a perfect square, sqrt(36+360*n)/6 = A090771 (numbers that are congruent to {1, 9} mod 10).

If a(n) == 324 mod 360 is a perfect square, sqrt(324+360*n)/6 = A063226 (numbers that are congruent to {3, 7} mod 10).

Examples

			1476 is in the sequence because 2*cos(1476°) = 2*cos(1476*Pi/180) = 1.61803398... = phi.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Cf. A001622, A019863.

Programs

Maple

***first program***
with(numtheory):err:=1/10^10:Digits:=20:for n from 1 to 20000 do:x:=evalf(2*cos(n*Pi/180)):ph:=evalf((1+sqrt(5)))/2:if abs(ph-x)
				

Mathematica

Select[Range[8000],2*Cos[# Degree]==GoldenRatio&] (* or *) LinearRecurrence[ {1,1,-1},{36,324,396},50] (* Harvey P. Dale, Aug 14 2015 *)

PARI

Vec(36*x*(x^2+8*x+1)/((x-1)^2*(x+1)) + O(x^100)) \\ Colin Barker, Feb 04 2014

Formula

a(n) = 18*(-5+3*(-1)^n+10*n). a(n) = a(n-1)+a(n-2)-a(n-3). G.f.: 36*x*(x^2+8*x+1) / ((x-1)^2*(x+1)). - Colin Barker, Feb 04 2014