OFFSET
1,9
COMMENTS
a(n) is the number of Wythoff AB-numbers from A003623 which are less than n.
a(n) is also the number of Wythoff A-pairs (two consecutive numbers which are both Wyhoff A-numbers) not exceeding n.
a(n) is also the number of Wythoff BA-numbers (including 2= B(A(1)) which however has Wythoff representation 0 for B(1)) not exceeding n-2. From the identity B(A(n)) = A(B(n))- 1.
LINKS
Martin Griffiths, A formula for an infinite family of Fibonacci-word sequences, Fib. Q., 56 (2018), 75-80.
FORMULA
a(n) = A(n+1) + A(n) - (3*n+1), with the Wythoff A-numbers A000201.
Note that no floor function definitions are necessary.
A(n) (which is as Beatty sequence also floor(n*phi), with phi=(1+sqrt(5))/2) can be defined from the rabbit sequence A005614(n-1), n>=1, which results from a substitution rule, via z(n) by A(n):= z(n-1) + n, B(n):= A(n) + n.
a(n) = floor(n/phi) - floor((1+n)/(1+phi)). - Frank Ruskey, Nov 30 2011
EXAMPLE
a(9) = 2 = A(10) + A(9) - (3*9+1) = 16 + 14 - 28.
a(9) = 2 = z(9) - z(8) - 9 = 6 + 5 - 9.
There are a(9)=2 AB-numbers <9, namely 3=A(B(1)) and 8=A(B(B(1)).
There are a(9)=2 A-pairs <=9, namely 3,4 and 8,9.
There are a(9)=2 BA-numbers <=7, namely 2 (see the comment above) and 7 = B(A(B(1))).
PROG
(Python)
from math import isqrt
def A192002(n): return (n+isqrt(m:=5*n**2)>>1)+(n+1+isqrt(m+10*n+5)>>1)-3*n-1 # Chai Wah Wu, Aug 10 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Jun 28 2011
STATUS
approved