A151915 Wythoff AAAA numbers.

1, 9, 14, 22, 30, 35, 43, 48, 56, 64, 69, 77, 85, 90, 98, 103, 111, 119, 124, 132, 137, 145, 153, 158, 166, 174, 179, 187, 192, 200, 208, 213, 221, 229, 234, 242, 247, 255, 263, 268, 276, 281, 289, 297, 302, 310, 318, 323, 331, 336, 344, 352, 357, 365, 370, 378

Offset: 1

Author

Benoit Cloitre, Apr 12 2008

Keywords

• nonn

Comments

In general Wythoff "AA...A" numbers, say A^(m)(n), with m A's, m>=1, are given by the formula: A^(m)(n) = F(m)*floor(n*Phi) + F(m-1)*n - F(m+1) + 1 where F are the Fibonacci numbers A000045.

Links

• A.H.M. Smeets, Table of n, a(n) for n = 1..20000
• Martin Griffiths, On a Matrix Arising from a Family of Iterated Self-Compositions, Journal of Integer Sequences, 18 (2015), #15.11.8.

Crossrefs

Cf. A001622, A000201 (Wythoff A numbers), A003622 (Wythoff AA numbers), A134859 (Wythoff AAA numbers).

Programs

Maple

A151915 := proc(n)
    g := (1+sqrt(5))/2 ;
    2*n-4+3*floor(n*g) ;
end proc:
seq(A151915(n),n=1..30) ; # R. J. Mathar, Jun 09 2018

Mathematica

a[n_] := 3 * Floor[n * GoldenRatio] + 2n - 4; Array[a, 100] (* Amiram Eldar, Dec 02 2018 *)

PARI

a(n)=3*floor(n/2*(1+sqrt(5)))+2*n-4

Python

from math import isqrt
def A151915(n): return (n-2<<1)+((m:=n+isqrt(5*n**2))&-2)+(m>>1) # Chai Wah Wu, Aug 10 2022

Formula

a(n) = A(A(A(A(n)))) where A(n) = A000201(n);

a(n) = 3*floor(n*phi) + 2*n - 4 = 3*A000201(n) + 2*n - 4, where phi = (1+sqrt(5))/2.