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

0,7

References

Martin Griffiths, A formula for an infinite family of Fibonacci-word sequences, Fib. Q., 56 (2018), 75-80.

Links

Harry J. Smith, Table of n, a(n) for n = 0..1000

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

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

Maple

A060144 := proc(n)
    (3+sqrt(5))/2 ;
    floor(n/%) ;
end proc:
seq(A060144(n), n=0..100) ; # R. J. Mathar, Jul 29 2021

Mathematica

Table[Floor[n/GoldenRatio^2], {n, 0, 100}] (* T. D. Noe, Dec 10 2011 *)

Prog

(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
(Haskell)
a060144 n = a060144_list !! n
a060144_list = 0 : 0 : scanl1 (+) a003849_list
-- Reinhard Zumkeller, Apr 07 2012
(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

Crossrefs

Cf. A001622, A005206, A006336.

Sequence in context: A225593 A057360 A057364 * A107347 A320063 A189717

Adjacent sequences: A060141 A060142 A060143 * A060145 A060146 A060147

Keyword

nonn

Author

Clark Kimberling, Mar 05 2001

Status

approved