A151915 - OEIS

%I #28 Mar 19 2024 22:34:46

%S 1,9,14,22,30,35,43,48,56,64,69,77,85,90,98,103,111,119,124,132,137,

%T 145,153,158,166,174,179,187,192,200,208,213,221,229,234,242,247,255,

%U 263,268,276,281,289,297,302,310,318,323,331,336,344,352,357,365,370,378

%N Wythoff AAAA numbers.

%C 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.

%H A.H.M. Smeets, <a href="/A151915/b151915.txt">Table of n, a(n) for n = 1..20000</a>

%H Martin Griffiths, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Griffiths/gr48.html">On a Matrix Arising from a Family of Iterated Self-Compositions</a>, Journal of Integer Sequences, 18 (2015), #15.11.8.

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

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

%p A151915 := proc(n)

%p     g := (1+sqrt(5))/2 ;

%p     2*n-4+3*floor(n*g) ;

%p end proc:

%p seq(A151915(n),n=1..30) ; # _R. J. Mathar_, Jun 09 2018

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

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

%o (Python)

%o from math import isqrt

%o def A151915(n): return (n-2<<1)+((m:=n+isqrt(5*n**2))&-2)+(m>>1) # _Chai Wah Wu_, Aug 10 2022

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

%K nonn

%O 1,2

%A _Benoit Cloitre_, Apr 12 2008