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.
3, 6, 11, 14, 19, 24, 27, 32, 35, 40, 45, 48, 53, 58, 61, 66, 69, 74, 79, 82, 87, 90, 95, 100, 103, 108, 113, 116, 121, 124, 129, 134, 137, 142, 147, 150, 155, 158, 163, 168, 171, 176, 179, 184, 189, 192, 197, 202, 205, 210, 213, 218, 223, 226, 231, 234, 239
Offset: 0
References
- Clark Kimberling, Stolarsky interspersions, Ars Combinatoria 39 (1995), 129-138.
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- 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.
- Aviezri S. Fraenkel, Recent results and questions in combinatorial game complexities, Theoretical Computer Science, vol. 249, no. 2 (2000), 265-288.
- Aviezri S. Fraenkel, Arrays, numeration systems and Frankenstein games, Theoret. Comput. Sci. 282 (2002), 271-284; preprint.
- Clark Kimberling, The first column of an interspersion, The Fibonacci Quarterly 32 (1994), 301-315.
Crossrefs
Cf. A007066.
Programs
-
Maple
A001950 := proc(n) local phi; phi := (1+sqrt(5))/2 ; floor(n*phi^2) ; end proc: A000201 := proc(n) local phi; phi := (1+sqrt(5))/2 ; floor(n*phi) ; end proc: A047924 := proc(n) 1+A001950(1+A000201(n)) ; end proc: # R. J. Mathar, Mar 20 2013
-
Mathematica
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 *) -
Python
from mpmath import * mp.dps=100 import math def A(n): return int(math.floor(n*phi)) def B(n): return int(math.floor(n*phi**2)) def a(n): return B(A(n) + 1) + 1 # Indranil Ghosh, Apr 25 2017
-
Python
from math import isqrt 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
Extensions
More terms from Naohiro Nomoto, Jun 08 2001
New description from Aviezri S. Fraenkel, Aug 03 2007
Comments