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

1,2

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.

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.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A272141 A289548 A001198 * A100263 A371059 A155082

Adjacent sequences: A151912 A151913 A151914 * A151916 A151917 A151918

Keyword

nonn

Author

Benoit Cloitre, Apr 12 2008

Status

approved