A035612 - OEIS

%I #63 Nov 05 2023 11:35:38

%S 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 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 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 Horizontal para-Fibonacci sequence: says which column of Wythoff array (starting column count at 1) contains n.

%C 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 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 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 Reinhard Zumkeller, <a href="/A035612/b035612.txt">Table of n, a(n) for n = 1..10000</a>

%H Paul Curtz, <a href="/A035612/a035612.txt">Comments on A035612, Jan 25 2016</a>.

%H 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 N. J. A. Sloane, <a href="/classic.html#WYTH">Classic Sequences</a>.

%F The segment between the first M and the first M+1 is given by the segment before the first M-1.

%F a(A022342(n)) > 1; a(A026274(n) + 1) = 1. - _Reinhard Zumkeller_, Jul 20 2015

%F 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 From _Rachel Chaiser_, Aug 18 2017: (Start)

%F 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 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 After the first 6 we see "1 2 3 1 4 1 2" then 7.

%t 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 (Haskell)

%o a035612 = a007814 . a022340

%o -- _Reinhard Zumkeller_, Jul 20 2015, Mar 10 2013

%Y Cf. A019586, A035513, A035614.

%Y Cf. A000045, A003603.

%Y Cf. A003714, A007814, A022340, A022342, A026274.

%Y Cf. A000012, A000027, A001045, A001610, A003622, A023548, A255671, A268034.

%Y Cf. A001622, A104457, A132338.

%K nonn,nice,easy

%O 1,2

%A _J. H. Conway_ and _N. J. A. Sloane_

%E Formula corrected by _Tom Edgar_, Jul 09 2018