The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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 (list; graph; refs; listen; history; text; internal format)
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
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]
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
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 29 04:26 EDT 2024. Contains 372921 sequences. (Running on oeis4.)