This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A101642 #54 Mar 29 2025 12:17:25
%S A101642 8,13,21,29,34,42,47,55,63,68,76,84,89,97,102,110,118,123,131,136,144,
%T A101642 152,157,165,173,178,186,191,199,207,212,220,228,233,241,246,254,262,
%U A101642 267,275,280,288,296,301,309,317,322,330,335,343,351,356,364,369,377
%N A101642 a(n) = Knuth's Fibonacci (or circle) product "3 o n".
%C A101642 Let phi be the golden ratio. Using _Fred Lunnon_'s formula in A101330 for Knuth's circle product, and the fact that phi^{-2} = 2-phi, plus [-x] = -[x]-1 for non-integer x, one obtains the formula below, expressing this sequence in terms of the lower Wythoff sequence. It follows in particular that the sequence of first differences 5,8,8,5,8,5,8,8,5,8,... of this sequence is the Fibonacci word A003849 on the alphabet {8,5}, shifted by 1. - _Michel Dekking_, Dec 23 2019
%C A101642 Also numbers with suffix string 0000, when written in Zeckendorf representation. - _A.H.M. Smeets_, Mar 20 2024
%C A101642 The asymptotic density of this sequence is 1/phi^4 = A094214^4 = 0.145898... . - _Amiram Eldar_, Mar 24 2025
%H A101642 A.H.M. Smeets, <a href="/A101642/b101642.txt">Table of n, a(n) for n = 1..20000</a>
%H A101642 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. 5.
%H A101642 W. F. Lunnon, <a href="/A101330/a101330.txt">Proof of formula</a>, 2008.
%F A101642 From _Michel Dekking_, Dec 23 2019: (Start)
%F A101642 a(n) = 3*A000201(n+1) + 2n - 3.
%F A101642 a(n) = A101345(n) + A000201(n+1) + n + 1. (End)
%t A101642 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[z1]}, {j, Length[z2]}]]; (* _Robert G. Wilson v_, Feb 04 2005 *)
%t A101642 Table[ kfp[3, n], {n, 50}] (* _Robert G. Wilson v_, Feb 04 2005 *)
%t A101642 Array[3*Floor[(# + 1)*GoldenRatio] + 2*# - 3 &, 100] (* _Paolo Xausa_, Mar 23 2024 *)
%o A101642 (Python)
%o A101642 from math import isqrt
%o A101642 def A101642(n): return 3*(n+1+isqrt(5*(n+1)**2)>>1)+(n<<1)-3 # _Chai Wah Wu_, Aug 29 2022
%Y A101642 Third row of array in A101330.
%Y A101642 Cf. A101345 = Knuth's Fibonacci (or circle) product "2 o n".
%Y A101642 Cf. A000201, A003849, A094214.
%K A101642 nonn
%O A101642 1,1
%A A101642 _N. J. A. Sloane_, Jan 26 2005
%E A101642 More terms from _David Applegate_, Jan 26 2005
%E A101642 More terms from _Robert G. Wilson v_, Feb 04 2005