OFFSET
1,3
REFERENCES
C. G. Lekkerkerker, Voorstelling van natuurlijke getallen door een som van getallen van Fibonacci, Simon Stevin vol. 29, 1952, pages 190-195.
E. Zeckendorf, Représentation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas, Bulletin de la Société Royale des Sciences de Liège vol. 41 (1972) pages 179-182.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 129 terms from Indranil Ghosh)
Ron Knott, Fibonacci Bases.
EXAMPLE
Fibonacci base columns are ...,8,5,3,2,1 with column entries 0 or 1 and no two consecutive ones (the Zeckendorf representation) so that each n has a unique representation.
12 is in the sequence because 12 = 8 + 3 + 1 = 10101 base Fib; 14 = 13 + 1 = 100001 base Fib.
MATHEMATICA
zeck[n_Integer] := Block[{k = Ceiling[ Log[ GoldenRatio, n*Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[ fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k-- ]; FromDigits[fr]]; a = {}; Do[z = zeck[n]; If[ FromDigits[ Reverse[ IntegerDigits[z]]] == z, AppendTo[a, n]], {n, 1123}]; a (* Robert G. Wilson v, May 29 2004 *)
mirror[dig_, s_] := Join[dig, s, Reverse[dig]]; select[v_, mid_] := Select[v, Length[#] == 0 || Last[#] != mid &]; fib[dig_] := Plus @@ (dig * Fibonacci[Range[2, Length[dig] + 1]]); pals = Rest[IntegerDigits /@ FromDigits /@ Select[Tuples[{0, 1}, 7], SequenceCount[#, {1, 1}] == 0 &]]; Union@Join[{0, 1}, fib /@ Join[mirror[#, {}] & /@ (select[pals, 1]), mirror[#, {1}] & /@ (select[pals, 1]), mirror[#, {0}] & /@ pals]] (* Amiram Eldar, Jan 11 2020 *)
PROG
(Python)
from sympy import fibonacci
def a(n):
k=0
x=0
while n>0:
k=0
while fibonacci(k)<=n: k+=1
x+=10**(k - 3)
n-=fibonacci(k - 1)
return x
def ok(n):
x=str(a(n))
return x==x[::-1]
print([n for n in range(1101) if ok(n)]) # Indranil Ghosh, Jun 07 2017
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Ron Knott, May 25 2004
EXTENSIONS
More terms from Robert G. Wilson v, May 28 2004
Offset changed to 1 by Alois P. Heinz, Aug 02 2017
STATUS
approved