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
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).
Crossrefs
Programs
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Mathematica
Select[Range@ 150, Or[# < 1/4, # > 3/4] &@ FractionalPart[# GoldenRatio] &] (* Michael De Vlieger, Nov 15 2017 *)
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PARI
isok(n) = my(phi=(1+sqrt(5))/2); (frac(n*phi)<1/4) || (frac(n*phi)>3/4); \\ Michel Marcus, Nov 14 2017
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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)
Comments