cp's OEIS Frontend

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

%I A286710 #18 Feb 19 2019 18:22:16
%S A286710 7,16,39,54,97,120,134,246,282,304,340,376,631,688,723,780,837,872,
%T A286710 929,964,1631,1722,1778,1869,1960,2016,2107,2163,2254,2345,2401,2492,
%U A286710 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
%N A286710 Numbers n whose Zeckendorf representation is of the form ww, for w a nonempty block of digits.
%C A286710 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.
%H A286710 Robert Israel, <a href="/A286710/b286710.txt">Table of n, a(n) for n = 1..10000</a>
%F A286710 a(A000045(n)) = A000045(n+1) + A000045(2n+1) for n >= 2. - _Robert Israel_, Feb 19 2019
%e A286710 The representation of 7 is 1010, which is of the form ww with w = 10.
%p A286710 F:= [seq(combinat:-fibonacci(i),i=2..21)]:
%p A286710 ext:= proc(L)
%p A286710   if L[2] = 0 then [0,op(L)], [0,1,op(L[2..-1])]
%p A286710   else [0,op(L)]
%p A286710   fi
%p A286710 end proc:
%p A286710 build:= proc(L) local i,k;
%p A286710   k:= nops(L);
%p A286710   add((F[i]+F[k+i])*L[i],i=1..k)
%p A286710 end proc:
%p A286710 R[2]:= [[0,1]]:
%p A286710 for i from 3 to 10 do R[i]:= map(ext,R[i-1]) od:
%p A286710 map(build, [seq(op(R[i]),i=2..10)]); # _Robert Israel_, Feb 19 2019
%t A286710 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 *)
%Y A286710 Cf. A000045, A014417, A094202 (the same sequence, but for palindromes).
%K A286710 nonn,base
%O A286710 1,1
%A A286710 _Jeffrey Shallit_, May 13 2017