A221158 -id:A221158 - cp's OEIS frontend

A222296 Irregular triangle read by rows: row n lists the Fibonacci numbers with exactly n 1's in their binary representation.

0, 1, 1, 2, 8, 3, 5, 34, 144, 13, 21

Offset: 0

T. D. Noe, Feb 22 2013

Besides those listed in Example section, there are no additional terms with small number of 1's in the first 10^12 Fibonacci numbers. In particular, if A000120(Fibonacci(n)) < 100, then n <= 319 or n > 10^12. - Charles R Greathouse IV, Mar 06 2014

For the theorem about S-units that Noam Elkies quotes (in the MathOverflow link), see Chapter 1 of Storey-Tijdemann, 1986. - N. J. A. Sloane, Jan 28 2017

			The irregular table begins
{0},
{1, 1, 2, 8},
{3, 5, 34, 144},
{13, 21, ...}.
It is conjectured that the previous (n=3) row is complete, and that the subsequent rows are:
{89, 610, 2584},
{55, 233, 4181},
{377, 10946, 46368, 75025},
{1597},
{987, 6765, 17711, 832040},
{121393, 2178309},
{39088169},
{28657, 196418, 317811, 1346269, 9227465},
{514229, 5702887, 14930352, 63245986, 4807526976},
{3524578, 2971215073}
...

T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge Tracts in Mathematics, 1986.

Charles Greathouse and Noam D. Elkies, Hamming weight of Fibonacci numbers, MathOverflow, 2014
Charles Greathouse and Noam D. Elkies, Hamming weight of Fibonacci numbers, MathOverflow, 2014 [Cached copy of three screen shots]
T. D. Noe, Conjectured values for rows n = 0..500 of irregular triangle, flattened
David Terr, On the sums of digits of Fibonacci numbers, Fibonacci Quarterly 34, Aug. 1996, pp. 349-355.

Cf. A004685 (Fibonacci numbers in binary), A221158 (weight 2), A222295, A222601, A222602, A222757, A222758.

f = Fibonacci[Range[0,100]]; Table[Select[f, Total[IntegerDigits[#, 2]] == n &], {n, 0, 20}]

row(n)=my(k=-1,t); while(1,t=fibonacci(k++); if(hammingweight(t)==n, print1(t", "))) \\ Charles R Greathouse IV, Mar 04 2014

a(9)-a(10) from Noam D. Elkies, via Charles R Greathouse IV, Mar 04 2014
Truncated to established terms by Max Alekseyev, May 13 2014
Edited by Max Alekseyev, Sep 08 2016

A222295 Conjectured number of Fibonacci numbers with exactly n bits set in their binary representation.

Original entry on oeis.org

1, 4, 4, 2, 3, 3, 4, 1, 4, 2, 1, 5, 5, 2, 2, 2, 5, 4, 3, 2, 2, 2, 3, 5, 3, 3, 2, 4, 2, 1, 4, 3, 2, 3, 3, 1, 6, 3, 2, 3, 3, 4, 4, 5, 0, 0, 3, 3, 2, 2, 5, 4, 3, 1, 5, 2, 2, 2, 5, 7, 3, 0, 0, 1, 2, 7, 3, 3, 2, 4, 3, 1, 2, 4, 4, 2, 0, 3, 1, 3, 7, 3, 4, 1, 3, 4, 3

Offset: 0

T. D. Noe, Feb 22 2013

			We set a(1) = 4 because Fib(1) = 1, Fib(2) = 1, Fib(3) = 2, and Fib(6) = 8.

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A004685 (Fibonacci numbers in binary), A221158 (two bits set), A222296.
Cf. A011373 (number of bits set in each Fibonacci number).
Cf. A222601, A222602, A222757, A222758.

f = Fibonacci[Range[0,500]]; Table[Length[Select[f, Total[IntegerDigits[#, 2]] == n &]], {n, 0, 87}]

A214853 Fibonacci numbers with only one 0 in the binary representation.

Original entry on oeis.org

0, 2, 5, 13, 55

Offset: 1

Alex Ratushnyak, Mar 08 2013

Conjecture: the sequence is finite.
No more terms below 2*10^301. - Matthew House, Sep 06 2015
No more terms below 10^162809483. (This number could easily be raised. Of the Fibonacci numbers less than 2^32 -- i.e., F(0) through F(47) -- F(10)=55 is the largest that has only one 0 in its binary representation, and of those not less than 2^32, the smallest one whose 32 least significant bits include fewer than 2 zero bits is Fibonacci(779038816), which exceeds 10^162809483.) - Jon E. Schoenfield, Sep 07 2015

			55 is 110111 in binary, thus 55 is in the sequence.

Cf. A004685, A221158.

Intersection of A030130 and A000045.

Select[Fibonacci@ Range[0, 120], Last@ DigitCount[#, 2] == 1 &] (* Michael De Vlieger, Sep 07 2015 *)

def count0(x):
    c = 0
    while x:
        c+= 1 - (x&1)
        if c>1:
            return 2
        x>>=1
    return c
prpr, prev = 0,1
TOP = 1<<12
print(0, end=',')
for i in range(1,TOP):
    if count0(prpr)==1:
        print(prpr, end=',')
    if (i&4095)==0:
        print('.', end=',')
    prpr, prev = prev, prpr+prev

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A222296 Irregular triangle read by rows: row n lists the Fibonacci numbers with exactly n 1's in their binary representation.

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A222295 Conjectured number of Fibonacci numbers with exactly n bits set in their binary representation.

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A214853 Fibonacci numbers with only one 0 in the binary representation.

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