A060144 - OEIS

%I #63 Jan 05 2025 19:51:36

%S 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,

%T 12,12,12,13,13,14,14,14,15,15,16,16,16,17,17,17,18,18,19,19,19,20,20,

%U 21,21,21,22,22,22,23,23,24,24,24,25,25,25,26,26,27,27,27,28,28,29,29

%N a(n) = floor(n/(1+tau)), or equivalently floor(n/(tau)^2), where tau is the golden ratio (A001622).

%H Harry J. Smith, <a href="/A060144/b060144.txt">Table of n, a(n) for n = 0..1000</a>

%H Martin Griffiths, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/56-1/GriffithsmgFibWordSeq121517.pdf">A formula for an infinite family of Fibonacci-word sequences</a>, Fib. Q., 56 (2018), 75-80.

%H D. R. Hofstadter, <a href="/A006336/a006336_1.pdf">Eta-Lore</a> [With permission]

%H D. R. Hofstadter, <a href="/A006336/a006336_2.pdf">Pi-Mu Sequences</a> [With permission]

%H D. R. Hofstadter and N. J. A. Sloane, <a href="/A006336/a006336.pdf">Correspondence, 1977 and 1991</a> (On page 4 of DRH letter, v[n] = A006336, a[n] = A060144[n+1]). - _N. J. A. Sloane_, Oct 25 2014

%F For n>0, a(n)=n reduced modulo A005206(n). - _Benoit Cloitre_, Jan 01 2003

%F 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

%F a(n+1) = n - A005206(n). - _Reinhard Zumkeller_, Apr 07 2012

%F a(n) = floor(n*A132338). - _R. J. Mathar_, Jul 29 2021

%p A060144 := proc(n)

%p     (3+sqrt(5))/2 ;

%p     floor(n/%) ;

%p end proc:

%p seq(A060144(n),n=0..100) ; # _R. J. Mathar_, Jul 29 2021

%t Table[Floor[n/GoldenRatio^2], {n, 0, 100}] (* _T. D. Noe_, Dec 10 2011 *)

%o (PARI)

%o { 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

%o (Haskell)

%o a060144 n = a060144_list !! n

%o a060144_list = 0 : 0 : scanl1 (+) a003849_list

%o -- _Reinhard Zumkeller_, Apr 07 2012

%o (Python)

%o from math import isqrt

%o def A060144(n): return (n<<1)-1-(n+isqrt(5*n**2)>>1) if n else 0 # _Chai Wah Wu_, Aug 09 2022

%Y Cf. A001622, A005206, A006336.

%K nonn,changed

%O 0,7

%A _Clark Kimberling_, Mar 05 2001