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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. 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 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

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