A295085 Numbers k such that {k*phi} < 0.25 or {k*phi} > 0.75, where phi is the golden ratio (1 + sqrt(5))/2 and { } denotes fractional part.

2, 3, 5, 8, 10, 11, 13, 16, 18, 21, 23, 24, 26, 29, 31, 32, 34, 36, 37, 39, 42, 44, 45, 47, 50, 52, 53, 55, 57, 58, 60, 63, 65, 66, 68, 71, 73, 76, 78, 79, 81, 84, 86, 87, 89, 91, 92, 94, 97, 99, 100, 102, 105, 107, 110, 112, 113, 115, 118, 120, 121, 123, 126, 128, 131, 133, 134, 136, 139, 141, 142, 144, 146

Offset

0,1

Comments

Numbers k such that k rotations by the golden angle yields a result between -Pi/2 and Pi/2 radians.

Links

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Burghard Herrmann, How integer sequences find their way into areas outside pure mathematics, The Fibonacci Quarterly (2019) Vol. 57, No. 5, 67-71.

P. Prusinkiewicz and A. Lindenmayer, Chapter 4, Phyllotaxis, The Algorithmic Beauty of Plants (1990).

Mathematica

Select[Range@ 150, Or[# < 1/4, # > 3/4] &@ FractionalPart[# GoldenRatio] &] (* Michael De Vlieger, Nov 15 2017 *)

Prog

(R) Phi=(sqrt(5)+1)/2 # Golden ratio
fp=function(x) x-floor(x) # fractional part
M=200
alpha=fp((1:M)*Phi) # angles in turn
PF=c(); PB=c() # Phyllotaxis front and back
for (i in 1:M) if ((alpha[i]>0.25)*(alpha[i]<0.75)) PB=c(PB, i) else PF=c(PF, i)
(PARI) isok(n) = my(phi=(1+sqrt(5))/2); (frac(n*phi)<1/4) || (frac(n*phi)>3/4); \\ Michel Marcus, Nov 14 2017

Crossrefs

Complement of A190250 (as has been proved), thus, intertwining of A190249 and A190251.

Cf. A001622.

Sequence in context: A028840 A189143 A047605 * A153000 A222172 A326379

Adjacent sequences: A295082 A295083 A295084 * A295086 A295087 A295088

Keyword

nonn,easy

Author

Burghard Herrmann, Nov 14 2017

Status

approved