A060144 a(n) = floor(n/(1+tau)), or equivalently floor(n/(tau)^2), where tau is the golden ratio (A001622).
0, 0, 0, 1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 14, 14, 14, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 19, 19, 19, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 24, 24, 24, 25, 25, 25, 26, 26, 27, 27, 27, 28, 28, 29, 29
Offset: 0
Keywords
Links
- Harry J. Smith, Table of n, a(n) for n = 0..1000
- Martin Griffiths, A formula for an infinite family of Fibonacci-word sequences, Fib. Q., 56 (2018), 75-80.
- D. R. Hofstadter, Eta-Lore [With permission]
- D. R. Hofstadter, Pi-Mu Sequences [With permission]
- D. R. Hofstadter and N. J. A. Sloane, Correspondence, 1977 and 1991 (On page 4 of DRH letter, v[n] = A006336, a[n] = A060144[n+1]). - _N. J. A. Sloane_, Oct 25 2014
- Johan Kok, Integer sequences with conjectured relation with certain graph parameters of the family of linear Jaco graphs, arXiv:2507.16500 [math.CO], 2025. See p. 4.
Programs
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Haskell
a060144 n = a060144_list !! n a060144_list = 0 : 0 : scanl1 (+) a003849_list -- Reinhard Zumkeller, Apr 07 2012
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Maple
A060144 := proc(n) (3+sqrt(5))/2 ; floor(n/%) ; end proc: seq(A060144(n),n=0..100) ; # R. J. Mathar, Jul 29 2021
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Mathematica
Table[Floor[n/GoldenRatio^2], {n, 0, 100}] (* T. D. Noe, Dec 10 2011 *) -
PARI
{ default(realprecision, 10); f=2/(sqrt(5) + 3); for (n=0, 1000, write("b060144.txt", n, " ", floor(n*f)); ) } \\ Harry J. Smith, Jul 02 2009 -
Python
from math import isqrt def A060144(n): return (n<<1)-1-(n+isqrt(5*n**2)>>1) if n else 0 # Chai Wah Wu, Aug 09 2022
Formula
For n>0, a(n)=n reduced modulo A005206(n). - Benoit Cloitre, Jan 01 2003
Let n' = n-1. Above formula is better as a(n') = n'-A005206(n'). Also a(n') = A005206(A005206(n'-1)). Also a(n'+1) = n'-a(n')-a(n'-a(n')), with a(0) = 0. - Frank Ruskey, Dec 09 2011
a(n+1) = n - A005206(n). - Reinhard Zumkeller, Apr 07 2012
a(n) = floor(n*A132338). - R. J. Mathar, Jul 29 2021
Comments