A352677 Binary expansion of largest number whose expansions in base 2 and in base phi = (1+sqrt(5))/2 are identical.

1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0

Offset

2

Links

Table of n, a(n) for n=2..106.

Chittaranjan Pardeshi, Python program

Formula

a(2)*phi + a(1) + a(0)/phi + ... + a(n)/phi^(1-n) + ... = a(2)*2 + a(1) + a(0)/2 + ... + a(n)/2^(1-n) + ....

A binary identical number is a number which shares the same digits in two different bases, b1 and b2, where 1 < b1 < b2 <= 2.

Here b2 = 2 and b1 = phi and initial digits "11." are the largest possible.

Algorithm:

1. set s1 = b1+1

2. set s2 = b2+1

3. output 1,1 # two digits above radix point

4. set i = 0

5. while s1 < s2:

6. i = i - 1

7. if s1 + b1^i <= s2 + b2^i:

8. s1 += b1^i

9. s2 += b2^i

10. output 1

11. else:

12. output 0

Example

11.11100001000100010100100010101000000010..., which in both binary and base phi is decimal 3.87916998...

Prog

(PARI)
f(b1, b2, len) = {
    my(d=0, c);
    vector(len, i,
        if((c=d+b1^(2-i)-b2^(2-i)) <= 0,
            d = c;
            1,
            0))
};
print(f(quadgen(5), 2, 1000));

Crossrefs

Cf. A001622 (phi).

Cf. A355328 (decimal + 1/2 = binary).

Sequence in context: A014064 A014119 A084052 * A130630 A059318 A071029

Adjacent sequences: A352674 A352675 A352676 * A352678 A352679 A352680

Keyword

nonn,cons,base

Author

Chittaranjan Pardeshi, Mar 27 2022

Status

approved