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A286710
Numbers n whose Zeckendorf representation is of the form ww, for w a nonempty block of digits.
1
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
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
KEYWORD
nonn,base
AUTHOR
Jeffrey Shallit, May 13 2017
STATUS
approved