OFFSET
1,1
COMMENTS
Numbers whose Zeckendorf representation ends with 000. - Benoit Cloitre, Jan 11 2014
The asymptotic density of this sequence is sqrt(5)-2. - Amiram Eldar, Mar 21 2022
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
Donald E. Knuth, Fibonacci multiplication, Appl. Math. Lett., Vol. 1, No. 1 (1988), pp. 57-60.
FORMULA
a(n) = floor(phi^3*(n+1)) - 3 - floor(2*phi*(n+1)) + 2*floor(phi*(n+1)) where phi = (1+sqrt(5))/2. - Benoit Cloitre, Jan 11 2014
MATHEMATICA
zeck[n_Integer] := Block[{k = Ceiling[ Log[ GoldenRatio, n * Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k-- ]; FromDigits[fr]]; kfp[n_, m_] := Block[{y = Reverse[ IntegerDigits[ zeck[ n]]], z = Reverse[ IntegerDigits[ zeck[ m]]]}, Sum[ y[[i]] * z[[j]] * Fibonacci[i + j + 2], {i, Length[y]}, {j, Length[z]}]];
Table[kfp[2, n], {n, 60}] (* Robert G. Wilson v, Feb 04 2005 *)
With[{r = Map[Fibonacci, Range[2, 14]]}, Rest[-1 + Position[#, _Integer][[All, 1]]] &@ Table[1/1000 * Total@ Map[FromDigits@ PadRight[{1}, Flatten@ #] &@ Reverse@ Position[r, #] &, Abs@ Differences@ NestWhileList[Function[k, k - SelectFirst[Reverse@ r, # < k &]], n + 1, # > 1 &]], {n, 0, 250}]] (* Michael De Vlieger, Jun 08 2017 *)
Array[2*Floor[(#+1)*GoldenRatio]+#-2 &, 100] (* Paolo Xausa, Mar 20 2024 *)
PROG
(Python)
from sympy import fibonacci
def a(n):
k=0
x=0
while n>0:
k=0
while fibonacci(k)<=n: k+=1
x+=10**(k - 3)
n-=fibonacci(k - 1)
return x
def ok(n): return 1 if str(a(n))[-3:]=="000" else 0 # Indranil Ghosh, Jun 08 2017
(Python)
from math import isqrt
def A101345(n): return (n+1+isqrt(5*(n+1)**2)&-2)+n-2 # Chai Wah Wu, Aug 29 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jan 26 2005
EXTENSIONS
More terms from David Applegate, Jan 26 2005
More terms from Robert G. Wilson v, Feb 04 2005
STATUS
approved