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%I #17 Jun 17 2017 03:09:44
%S 36,324,396,684,756,1044,1116,1404,1476,1764,1836,2124,2196,2484,2556,
%T 2844,2916,3204,3276,3564,3636,3924,3996,4284,4356,4644,4716,5004,
%U 5076,5364,5436,5724,5796,6084,6156,6444,6516,6804,6876,7164,7236,7524,7596,7884
%N Angles n expressed in degrees such that 2*cos(n) = phi where phi is the golden ratio (A001622).
%C a(n) == 36, 324 mod 360 and a(n)/36 is congruent to {1,9} mod 10 (A090771).
%C See A019863 = half of the golden ratio (A001622) => a(1) = 90 - 54 degrees and a(2) = 360 - a(1) = 324 degrees.
%C 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:
%C 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).
%C 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).
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F 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
%e 1476 is in the sequence because 2*cos(1476°) = 2*cos(1476*Pi/180) = 1.61803398... = phi.
%p ***first program***
%p 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)<err then printf(`%d, `,n):else fi:od:
%p ***second program***
%p lst:={}:for n from 0 to 30 do:x:=36+n*360:y:=324+n*360:lst:=lst union {x} union {y}:od:print(lst):
%t Select[Range[8000],2*Cos[# Degree]==GoldenRatio&] (* or *) LinearRecurrence[ {1,1,-1},{36,324,396},50] (* _Harvey P. Dale_, Aug 14 2015 *)
%o (PARI) Vec(36*x*(x^2+8*x+1)/((x-1)^2*(x+1)) + O(x^100)) \\ _Colin Barker_, Feb 04 2014
%Y Cf. A001622, A019863.
%K nonn,easy
%O 1,1
%A _Michel Lagneau_, Feb 04 2014
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