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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 (list; graph; refs; listen; history; text; internal format)
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).

LINKS

Table of n, a(n) for n=1..44.

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

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

Cf. A001622, A019863.

Sequence in context: A232828 A092643 A045786 * A171586 A017594 A014800

Adjacent sequences:  A237125 A237126 A237127 * A237129 A237130 A237131

KEYWORD

nonn,easy

AUTHOR

Michel Lagneau, Feb 04 2014

STATUS

approved

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Last modified February 18 02:20 EST 2018. Contains 299297 sequences. (Running on oeis4.)