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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.

%I A222296 #65 Jan 05 2025 19:51:40
%S A222296 0,1,1,2,8,3,5,34,144,13,21
%N A222296 Irregular triangle read by rows: row n lists the Fibonacci numbers with exactly n 1's in their binary representation.
%C A222296 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
%C A222296 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
%D A222296 T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge Tracts in Mathematics, 1986.
%H A222296 Charles Greathouse and Noam D. Elkies, <a href="http://mathoverflow.net/questions/159142">Hamming weight of Fibonacci numbers</a>, MathOverflow, 2014
%H A222296 Charles Greathouse and Noam D. Elkies, <a href="/A222296/a222296.pdf">Hamming weight of Fibonacci numbers</a>, MathOverflow, 2014 [Cached copy of three screen shots]
%H A222296 T. D. Noe, <a href="/A222296/a222296.txt">Conjectured values for rows n = 0..500 of irregular triangle, flattened</a>
%H A222296 David Terr, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/34-4/terr.pdf">On the sums of digits of Fibonacci numbers,</a> Fibonacci Quarterly 34, Aug. 1996, pp. 349-355.
%e A222296 The irregular table begins
%e A222296 {0},
%e A222296 {1, 1, 2, 8},
%e A222296 {3, 5, 34, 144},
%e A222296 {13, 21, ...}.
%e A222296 It is conjectured that the previous (n=3) row is complete, and that the subsequent rows are:
%e A222296 {89, 610, 2584},
%e A222296 {55, 233, 4181},
%e A222296 {377, 10946, 46368, 75025},
%e A222296 {1597},
%e A222296 {987, 6765, 17711, 832040},
%e A222296 {121393, 2178309},
%e A222296 {39088169},
%e A222296 {28657, 196418, 317811, 1346269, 9227465},
%e A222296 {514229, 5702887, 14930352, 63245986, 4807526976},
%e A222296 {3524578, 2971215073}
%e A222296 ...
%t A222296 f = Fibonacci[Range[0,100]]; Table[Select[f, Total[IntegerDigits[#, 2]] == n &], {n, 0, 20}]
%o A222296 (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
%Y A222296 Cf. A004685 (Fibonacci numbers in binary), A221158 (weight 2), A222295, A222601, A222602, A222757, A222758.
%K A222296 nonn,base,tabf,more,hard
%O A222296 0,4
%A A222296 _T. D. Noe_, Feb 22 2013
%E A222296 a(9)-a(10) from _Noam D. Elkies_, via _Charles R Greathouse IV_, Mar 04 2014
%E A222296 Truncated to established terms by _Max Alekseyev_, May 13 2014
%E A222296 Edited by _Max Alekseyev_, Sep 08 2016