A192002 Counting sequence for Wythoff AB-numbers smaller than n.

0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 18, 18, 18, 19, 19, 19, 19, 19, 20, 20, 20, 21, 21, 21

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.

For the Wythoff representation of numbers see A189921 and A135817.

Links

Table of n, a(n) for n=1..90.

Martin Griffiths, A formula for an infinite family of Fibonacci-word sequences, Fib. Q., 56 (2018), 75-80.

Formula

a(n) = Sum_{j=0..n-1} A123740(j), with A123740(0)=0.

a(n) = A(n+1) + A(n) - (3*n+1), with the Wythoff A-numbers A000201.

a(n) = z(n) + z(n-1) - n, with z(n) = A005206(n) = A060143(n+1) which counts A-numbers <=n.

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

Sequence in context: A003059 A325678 A247189 * A263270 A337636 A011752

Adjacent sequences: A191999 A192000 A192001 * A192003 A192004 A192005

Keyword

nonn,easy,changed

Author

Wolfdieter Lang, Jun 28 2011

Status

approved