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

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

Offset: 1

Author

N. J. A. Sloane, Jan 26 2005

Keywords

• nonn

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
• Jon Asier Bárcena-Petisco, Luis Martínez, María Merino, Juan Manuel Montoya, and Antonio Vera-López, Fibonacci-like partitions and their associated piecewise-defined permutations, arXiv:2503.19696 [math.CO], 2025. See p. 4.
• Donald E. Knuth, Fibonacci multiplication, Appl. Math. Lett., Vol. 1, No. 1 (1988), pp. 57-60.

Crossrefs

Second row of array in A101330.

Cf. A000201, A014417, A101642, A095085.

Set-wise difference of A026274 - A035337.

Programs

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 *)

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

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

a(n) = 2*A000201(n+1) + n - 2. See the comments in A101642. - Michel Dekking, Dec 23 2019

Extensions

More terms from David Applegate, Jan 26 2005

More terms from Robert G. Wilson v, Feb 04 2005