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

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

2nd column of array in A038150.

Apart from the first term also the second column of A126714; see also A223025. - Casey Mongoven, Mar 11 2013

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