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A101345 a(n) = Knuth's Fibonacci (or circle) product "2 o n".

%I A101345 #45 Mar 29 2025 12:17:03
%S A101345 5,8,13,18,21,26,29,34,39,42,47,52,55,60,63,68,73,76,81,84,89,94,97,
%T A101345 102,107,110,115,118,123,128,131,136,141,144,149,152,157,162,165,170,
%U A101345 173,178,183,186,191,196,199,204,207,212,217,220,225,228,233,238,241,246
%N A101345 a(n) = Knuth's Fibonacci (or circle) product "2 o n".
%C A101345 Numbers whose Zeckendorf representation ends with 000. - _Benoit Cloitre_, Jan 11 2014
%C A101345 The asymptotic density of this sequence is sqrt(5)-2. - _Amiram Eldar_, Mar 21 2022
%H A101345 Amiram Eldar, <a href="/A101345/b101345.txt">Table of n, a(n) for n = 1..10000</a>
%H A101345 Jon Asier Bárcena-Petisco, Luis Martínez, María Merino, Juan Manuel Montoya, and Antonio Vera-López, <a href="https://arxiv.org/abs/2503.19696">Fibonacci-like partitions and their associated piecewise-defined permutations</a>, arXiv:2503.19696 [math.CO], 2025. See p. 4.
%H A101345 Donald E. Knuth, <a href="http://dx.doi.org/10.1016/0893-9659(88)90176-0">Fibonacci multiplication</a>, Appl. Math. Lett., Vol. 1, No. 1 (1988), pp. 57-60.
%F A101345 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
%F A101345 a(n) = 2*A000201(n+1) + n - 2. See the comments in A101642. - _Michel Dekking_, Dec 23 2019
%t A101345 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]}]];
%t A101345 Table[kfp[2, n], {n, 60}] (* _Robert G. Wilson v_, Feb 04 2005 *)
%t A101345 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 *)
%t A101345 Array[2*Floor[(#+1)*GoldenRatio]+#-2 &, 100] (* _Paolo Xausa_, Mar 20 2024 *)
%o A101345 (Python)
%o A101345 from sympy import fibonacci
%o A101345 def a(n):
%o A101345     k=0
%o A101345     x=0
%o A101345     while n>0:
%o A101345         k=0
%o A101345         while fibonacci(k)<=n: k+=1
%o A101345         x+=10**(k - 3)
%o A101345         n-=fibonacci(k - 1)
%o A101345     return x
%o A101345 def ok(n): return 1 if str(a(n))[-3:]=="000" else 0 # _Indranil Ghosh_, Jun 08 2017
%o A101345 (Python)
%o A101345 from math import isqrt
%o A101345 def A101345(n): return (n+1+isqrt(5*(n+1)**2)&-2)+n-2 # _Chai Wah Wu_, Aug 29 2022
%Y A101345 Second row of array in A101330.
%Y A101345 Cf. A000201, A014417, A101642, A095085.
%Y A101345 Set-wise difference of A026274 - A035337.
%K A101345 nonn
%O A101345 1,1
%A A101345 _N. J. A. Sloane_, Jan 26 2005
%E A101345 More terms from _David Applegate_, Jan 26 2005
%E A101345 More terms from _Robert G. Wilson v_, Feb 04 2005