|
%I M3935 N1619 #66 Oct 27 2023 19:31:19
%S 0,1,5,25,29,41,49,61,65,85,89,101,125,145,149,245,265,365,385,485,
%T 505,601,605,625,649,701,725,745,749,845,865,965,985,1105,1205,1249,
%U 1345,1445,1585,1685,1825,1925,2065,2165,2305,2405,2501,2545,2645,2785,2885
%N Numbers m such that Fibonacci(m) ends with m.
%C Conjecture: Other than 1 and 5, there is no m such that Fibonacci(m) in binary ends with m in binary. The conjecture holds up to m=50000. - _Ralf Stephan_, Aug 21 2006
%C The conjecture for binary numbers holds for m < 2^25. - _T. D. Noe_, May 14 2007
%C Conjecture is true. It is easy to prove (by induction on k) that if Fibonacci(m) ends with m in binary, then m == 0, 1, or 5 (mod 3*2^k) for any positive integer k, i.e., m must simply be equal to 0, 1, or 5. - _Max Alekseyev_, Jul 03 2009
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A000350/b000350.txt">Table of n, a(n) for n = 1..1034</a> (terms n = 1..803 from T. D. Noe)
%H G. R. Deily, <a href="https://www.fq.math.ca/Scanned/4-2/deily.pdf">Terminal Digit Coincidences Between Fibonacci Numbers and Their Indices</a>, The Fibonacci Quarterly, 4.2 (1966) 151.
%H M. Dunton and R. E. Grimm, <a href="http://www.fq.math.ca/Scanned/4-4/dunton.pdf">Fibonacci on Egyptian fractions</a>, Fib. Quart., 4 (1966), 339-354.
%H D. A. Lind, <a href="https://www.fq.math.ca/Scanned/5-2/lind.pdf">Extended Computations of Terminal Digit Coincidences</a>, Fibonacci Quarterly, 5.2 April 1967 pp. 183-184.
%e Fibonacci(25) = 75025 ends with 25.
%t a=0;b=1;c=1;lst={}; Do[a=b;b=c;c=a+b;m=Floor[N[Log[10,n]]]+1; If[Mod[c,10^m]==n,AppendTo[lst,n]],{n,3,5000}]; Join[{0,1},lst] (* edited and changed by _Harvey P. Dale_, Sep 10 2011 *)
%t fnQ[n_]:=Mod[Fibonacci[n],10^IntegerLength[n]]==n; Select[Range[ 0,2900],fnQ] (* _Harvey P. Dale_, Nov 03 2012 *)
%o (Haskell)
%o import Data.List (isSuffixOf, elemIndices)
%o import Data.Function (on)
%o a000350 n = a000350_list !! (n-1)
%o a000350_list = elemIndices True $
%o zipWith (isSuffixOf `on` show) [0..] a000045_list
%o -- _Reinhard Zumkeller_, Apr 10 2012
%o (PARI) for(n=0,1e4,if(((Mod([1,1;1,0],10^#Str(n)))^n)[1,2]==n,print1(n", "))) \\ _Charles R Greathouse IV_, Apr 10 2012
%o (Python)
%o from sympy import fibonacci
%o [i for i in range(1000) if str(fibonacci(i))[-len(str(i)):]==str(i)] # _Nicholas Stefan Georgescu_, Feb 27 2023
%Y Cf. A000045, A050816, A038546, A052000, A023172.
%K nonn,base,easy,nice
%O 1,3
%A _N. J. A. Sloane_
|
