

A222296


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


5



0, 1, 1, 2, 8, 3, 5, 34, 144, 13, 21
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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 Sunits that Noam Elkies quotes (in the MathOverflow link), see Chapter 1 of StoreyTijdemann, 1986.  N. J. A. Sloane, Jan 28 2017


REFERENCES

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


LINKS



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



KEYWORD

nonn,base,tabf,more,hard


AUTHOR



EXTENSIONS



STATUS

approved



