cp's OEIS Frontend

A047924 a(n) = B_{A_n+1}+1, where A_n = floor(n*phi) = A000201(n), B_n = floor(n*phi^2) = A001950(n) and phi is the golden ratio.

%I A047924 #43 Mar 28 2025 10:28:13
%S A047924 3,6,11,14,19,24,27,32,35,40,45,48,53,58,61,66,69,74,79,82,87,90,95,
%T A047924 100,103,108,113,116,121,124,129,134,137,142,147,150,155,158,163,168,
%U A047924 171,176,179,184,189,192,197,202,205,210,213,218,223,226,231,234,239
%N A047924 a(n) = B_{A_n+1}+1, where A_n = floor(n*phi) = A000201(n), B_n = floor(n*phi^2) = A001950(n) and phi is the golden ratio.
%C A047924 2nd column of array in A038150.
%C A047924 Apart from the first term also the second column of A126714; see also A223025. - _Casey Mongoven_, Mar 11 2013
%D A047924 Clark Kimberling, Stolarsky interspersions, Ars Combinatoria 39 (1995), 129-138.
%H A047924 Vincenzo Librandi, <a href="/A047924/b047924.txt">Table of n, a(n) for n = 0..1000</a>
%H A047924 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 A047924 Aviezri S. Fraenkel, <a href="https://doi.org/10.1016/S0304-3975(00)00062-1">Recent results and questions in combinatorial game complexities</a>, Theoretical Computer Science, vol. 249, no. 2 (2000), 265-288.
%H A047924 Aviezri S. Fraenkel, <a href="https://doi.org/10.1016/S0304-3975(01)00070-6">Arrays, numeration systems and Frankenstein games</a>, Theoret. Comput. Sci. 282 (2002), 271-284; <a href="http://www.wisdom.weizmann.ac.il/~fraenkel/Papers/ans1.ps">preprint</a>.
%H A047924 Clark Kimberling, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/32-4/kimberling.pdf">The first column of an interspersion</a>, The Fibonacci Quarterly 32 (1994), 301-315.
%p A047924 A001950 := proc(n)
%p A047924         local phi;
%p A047924         phi := (1+sqrt(5))/2 ;
%p A047924         floor(n*phi^2) ;
%p A047924 end proc:
%p A047924 A000201 := proc(n)
%p A047924         local phi;
%p A047924         phi := (1+sqrt(5))/2 ;
%p A047924         floor(n*phi) ;
%p A047924 end proc:
%p A047924 A047924 := proc(n)
%p A047924         1+A001950(1+A000201(n)) ;
%p A047924 end proc: # _R. J. Mathar_, Mar 20 2013
%t A047924 A[n_] := Floor[n*GoldenRatio]; B[n_] := Floor[n*GoldenRatio^2]; a[n_] := B[A[n]+1]+1; Table[a[n], {n, 0, 56}] (* _Jean-François Alcover_, Feb 11 2014 *)
%o A047924 (Python)
%o A047924 from mpmath import *
%o A047924 mp.dps=100
%o A047924 import math
%o A047924 def A(n): return int(math.floor(n*phi))
%o A047924 def B(n): return int(math.floor(n*phi**2))
%o A047924 def a(n): return B(A(n) + 1) + 1 # _Indranil Ghosh_, Apr 25 2017
%o A047924 (Python)
%o A047924 from math import isqrt
%o A047924 def A047924(n): return ((m:=(n+isqrt(5*n**2)>>1)+1)+isqrt(5*m**2)>>1)+m+1 # _Chai Wah Wu_, Aug 25 2022
%Y A047924 Cf. A007066.
%K A047924 nonn,nice,easy
%O A047924 0,1
%A A047924 _N. J. A. Sloane_
%E A047924 More terms from _Naohiro Nomoto_, Jun 08 2001
%E A047924 New description from _Aviezri S. Fraenkel_, Aug 03 2007