A286710 Numbers n whose Zeckendorf representation is of the form ww, for w a nonempty block of digits.

7, 16, 39, 54, 97, 120, 134, 246, 282, 304, 340, 376, 631, 688, 723, 780, 837, 872, 929, 964, 1631, 1722, 1778, 1869, 1960, 2016, 2107, 2163, 2254, 2345, 2401, 2492, 2583, 4236, 4382, 4472, 4618, 4764, 4854, 5000, 5090, 5236, 5382, 5472, 5618, 5764, 5854, 6000, 6090, 6236, 6382, 6472, 6618, 6708, 11035, 11270, 11415

Offset

1,1

Comments

The Zeckendorf representation of an integer n expresses n as a sum of non-adjacent Fibonacci numbers. It can be expressed as a word over {0,1} giving the coefficients, starting with the most significant digit.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

a(A000045(n)) = A000045(n+1) + A000045(2n+1) for n >= 2. - Robert Israel, Feb 19 2019

Example

The representation of 7 is 1010, which is of the form ww with w = 10.

Maple

F:= [seq(combinat:-fibonacci(i), i=2..21)]:
ext:= proc(L)
  if L[2] = 0 then [0, op(L)], [0, 1, op(L[2..-1])]
  else [0, op(L)]
  fi
end proc:
build:= proc(L) local i, k;
  k:= nops(L);
  add((F[i]+F[k+i])*L[i], i=1..k)
end proc:
R[2]:= [[0, 1]]:
for i from 3 to 10 do R[i]:= map(ext, R[i-1]) od:
map(build, [seq(op(R[i]), i=2..10)]); # Robert Israel, Feb 19 2019

Mathematica

Reap[Do[ w = IntegerDigits[k, 2]; p = 1 + Flatten@ Position[ Reverse@ Join[w, w], 1]; If[ Min@ Differences@ p > 1, Sow@ Total@ Fibonacci@ p], {k, 2^10 - 1}]][[2, 1]] (* Giovanni Resta, May 13 2017 *)

Crossrefs

Cf. A000045, A014417, A094202 (the same sequence, but for palindromes).

Sequence in context: A278945 A327628 A169877 * A036834 A020941 A318750

Adjacent sequences: A286707 A286708 A286709 * A286711 A286712 A286713

Keyword

nonn,base

Author

Jeffrey Shallit, May 13 2017

Status

approved