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

Author

Jeffrey Shallit, May 13 2017

Keywords

• nonn
• base

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.

Examples

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

Links

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

Crossrefs

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

Programs

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 *)

Formula

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