login
A060144
a(n) = floor(n/(1+tau)), or equivalently floor(n/(tau)^2), where tau is the golden ratio (A001622).
15
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
LINKS
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
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
KEYWORD
nonn
AUTHOR
Clark Kimberling, Mar 05 2001
STATUS
approved