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A101345
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a(n) = Knuth's Fibonacci (or circle) product "2 o n".
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6
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5, 8, 13, 18, 21, 26, 29, 34, 39, 42, 47, 52, 55, 60, 63, 68, 73, 76, 81, 84, 89, 94, 97, 102, 107, 110, 115, 118, 123, 128, 131, 136, 141, 144, 149, 152, 157, 162, 165, 170, 173, 178, 183, 186, 191, 196, 199, 204, 207, 212, 217, 220, 225, 228, 233, 238, 241, 246
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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FORMULA
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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
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MATHEMATICA
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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]}]];
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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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