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A237128
Angles n expressed in degrees such that 2*cos(n) = phi where phi is the golden ratio (A001622).
0
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
1,1
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).
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
EXAMPLE
1476 is in the sequence because 2*cos(1476°) = 2*cos(1476*Pi/180) = 1.61803398... = phi.
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)<err then printf(`%d, `, n):else fi:od:
***second program***
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):
MATHEMATICA
Select[Range[8000], 2*Cos[# Degree]==GoldenRatio&] (* or *) LinearRecurrence[ {1, 1, -1}, {36, 324, 396}, 50] (* Harvey P. Dale, Aug 14 2015 *)
PROG
(PARI) Vec(36*x*(x^2+8*x+1)/((x-1)^2*(x+1)) + O(x^100)) \\ Colin Barker, Feb 04 2014
CROSSREFS
Sequence in context: A232828 A092643 A045786 * A171586 A017594 A014800
KEYWORD
nonn,easy
AUTHOR
Michel Lagneau, Feb 04 2014
STATUS
approved