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

Author

T. D. Noe, Feb 22 2013

Keywords

• nonn
• base
• tabf
• more
• hard

Comments

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

Examples

			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}
...
		

References

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

Links

• 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.

Crossrefs

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

Programs

Mathematica

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

PARI

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

Extensions

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