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

0,4

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

References

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

Links

Table of n, a(n) for n=0..10.

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.

Example

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}
...

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A243340 A173823 A278115 * A258223 A154928 A201394

Adjacent sequences: A222293 A222294 A222295 * A222297 A222298 A222299

Keyword

nonn,base,tabf,more,hard

Author

T. D. Noe, Feb 22 2013

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

Status

approved