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A000350
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Numbers n such that Fibonacci(n) ends with n.
(Formerly M3935 N1619)
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7
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0, 1, 5, 25, 29, 41, 49, 61, 65, 85, 89, 101, 125, 145, 149, 245, 265, 365, 385, 485, 505, 601, 605, 625, 649, 701, 725, 745, 749, 845, 865, 965, 985, 1105, 1205, 1249, 1345, 1445, 1585, 1685, 1825, 1925, 2065, 2165, 2305, 2405, 2501, 2545, 2645, 2785, 2885
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,3
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COMMENTS
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Conjecture: Other than 1 and 5, there is no n such that Fibonacci(n) in binary ends with n in binary. The conjecture holds up to n=50000. - Ralf Stephan, Aug 21 2006
The conjecture for binary numbers holds for n < 2^25. - T. D. Noe, May 14 2007
Conjecture is true. It is easy to prove (by induction on k) that if F(n) ends with n in binary, then n == 0, 1, or 5 modulo 3*2^k for any positive integer k, i.e., n must simply be equal to 0, 1, or 5. - Max Alekseyev, Jul 03 2009
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REFERENCES
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N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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T. D. Noe and Alois P. Heinz, Table of n, a(n) for n = 1..1034 (terms n = 1..803 from T. D. Noe)
G. R. Deily, Terminal Digit Coincidences Between Fibonacci Numbers and Their Indices, The Fibonacci Quarterly, 4.2 (1966) 151.
M. Dunton and R. E. Grimm, Fibonacci on Egyptian fractions, Fib. Quart., 4 (1966), 339-354.
D. A. Lind, Extended Computations of Terminal Digit Coincidences, Fibonacci Quarterly, 5.2 April 1967 pp. 183-184.
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EXAMPLE
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Fibonacci(25) = 75025 ends with 25.
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MATHEMATICA
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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 *)
fnQ[n_]:=Mod[Fibonacci[n], 10^IntegerLength[n]]==n; Select[Range[ 0, 2900], fnQ] (* Harvey P. Dale, Nov 03 2012 *)
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PROG
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(Haskell)
import Data.List (isSuffixOf, elemIndices)
import Data.Function (on)
a000350 n = a000350_list !! (n-1)
a000350_list = elemIndices True $
zipWith (isSuffixOf `on` show) [0..] a000045_list
-- Reinhard Zumkeller, Apr 10 2012
(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
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CROSSREFS
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Cf. A000045, A050816, A038546, A052000, A023172.
Sequence in context: A036137 A070380 A068574 * A000221 A018612 A036127
Adjacent sequences: A000347 A000348 A000349 * A000351 A000352 A000353
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KEYWORD
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nonn,base,easy,nice
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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