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 A035612 #63 Nov 05 2023 11:35:38
%S A035612 1,2,3,1,4,1,2,5,1,2,3,1,6,1,2,3,1,4,1,2,7,1,2,3,1,4,1,2,5,1,2,3,1,8,
%T A035612 1,2,3,1,4,1,2,5,1,2,3,1,6,1,2,3,1,4,1,2,9,1,2,3,1,4,1,2,5,1,2,3,1,6,
%U A035612 1,2,3,1,4,1,2,7,1,2,3,1,4,1,2,5,1,2,3,1,10,1,2,3,1,4,1,2,5,1,2
%N A035612 Horizontal para-Fibonacci sequence: says which column of Wythoff array (starting column count at 1) contains n.
%C A035612 Ordinal transform of A003603. Removing all 1's from this sequence and decrementing the remaining numbers generates the original sequence. - _Franklin T. Adams-Watters_, Aug 10 2012
%C A035612 It can be shown that a(n) is the index of the smallest Fibonacci number used in the Zeckendorf representation of n, where f(0)=f(1)=1. - _Rachel Chaiser_, Aug 18 2017
%C A035612 The asymptotic density of the occurrences of k = 1, 2, ..., is (2-phi)/phi^(k-1), where phi is the golden ratio (A001622). The asymptotic mean of this sequence is 1 + phi (A104457). - _Amiram Eldar_, Nov 02 2023
%H A035612 Reinhard Zumkeller, <a href="/A035612/b035612.txt">Table of n, a(n) for n = 1..10000</a>
%H A035612 Paul Curtz, <a href="/A035612/a035612.txt">Comments on A035612, Jan 25 2016</a>.
%H A035612 N. J. A. Sloane, <a href="http://neilsloane.com/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).
%H A035612 N. J. A. Sloane, <a href="/classic.html#WYTH">Classic Sequences</a>.
%F A035612 The segment between the first M and the first M+1 is given by the segment before the first M-1.
%F A035612 a(A022342(n)) > 1; a(A026274(n) + 1) = 1. - _Reinhard Zumkeller_, Jul 20 2015
%F A035612 a(n) = v2(A022340(n)), where v2(n) = A007814(n), the dyadic valuation of n. - _Ralf Stephan_, Jun 20 2004. In other words, a(n) = A007814(A003714(n)) + 1, which is certainly true. - _Don Reble_, Nov 12 2005
%F A035612 From _Rachel Chaiser_, Aug 18 2017: (Start)
%F A035612 a(n) = a(p(n))+1 if n = b(p(n)) where p(n) = floor((n+2)/phi)-1 and b(n) = floor((n+1)*phi)-1 where phi=(1+sqrt(5))/2; a(n)=1 otherwise.
%F A035612 a(n) = 3 - n_1 + s_z(n-1) - s_z(n) + s_z(p(n-1)) - s_z(p(n)), where s_z(n) is the Zeckendorf sum of digits of n (A007895), and n_1 is the least significant digit in the Zeckendorf representation of n. (End)
%e A035612 After the first 6 we see "1 2 3 1 4 1 2" then 7.
%t A035612 f[1] = {1}; f[2] = {1, 2}; f[n_] := f[n] = Join[f[n-1], Most[f[n-2]], {n}]; f[11] (* _Jean-François Alcover_, Feb 22 2012 *)
%o A035612 (Haskell)
%o A035612 a035612 = a007814 . a022340
%o A035612 -- _Reinhard Zumkeller_, Jul 20 2015, Mar 10 2013
%Y A035612 Cf. A019586, A035513, A035614.
%Y A035612 Cf. A000045, A003603.
%Y A035612 Cf. A003714, A007814, A022340, A022342, A026274.
%Y A035612 Cf. A000012, A000027, A001045, A001610, A003622, A023548, A255671, A268034.
%Y A035612 Cf. A001622, A104457, A132338.
%K A035612 nonn,nice,easy
%O A035612 1,2
%A A035612 _J. H. Conway_ and _N. J. A. Sloane_
%E A035612 Formula corrected by _Tom Edgar_, Jul 09 2018