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

0,1

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

A. S. Fraenkel, Recent results and questions in combinatorial game complexities, Theoretical Computer Science, vol. 249, no. 2 (2000), 265-288.

A. 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.

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 *)

Prog

(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

Crossrefs

Cf. A007066.

Sequence in context: A246976 A189380 A047398 * A267519 A200182 A342173

Adjacent sequences: A047921 A047922 A047923 * A047925 A047926 A047927

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Extensions

More terms from Naohiro Nomoto, Jun 08 2001

New description from Aviezri S. Fraenkel, Aug 03 2007

Status

approved