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A377092 a(0) = 0, and for any n > 0, a(n) is the least integer (in absolute value) not yet in the sequence such that the absolute difference of a(n-1) and a(n) is a Fibonacci number (A000045); in case of a tie, preference is given to the positive value.

%I A377092 #34 Apr 19 2025 18:01:55
%S A377092 0,1,-1,2,3,-2,-3,-4,4,5,6,7,-6,-5,-7,-8,-9,-10,11,8,9,10,-11,-12,-13,
%T A377092 -14,-15,-16,-17,17,12,13,14,15,16,18,19,20,21,22,23,24,25,26,27,28,
%U A377092 -27,-19,-18,-20,-21,-22,-23,-24,-25,-26,-28,-29,-30,-31,-32,-33
%N A377092 a(0) = 0, and for any n > 0, a(n) is the least integer (in absolute value) not yet in the sequence such that the absolute difference of a(n-1) and a(n) is a Fibonacci number (A000045); in case of a tie, preference is given to the positive value.
%C A377092 Will every integer appear in the sequence?
%H A377092 Rémy Sigrist, <a href="/A377092/b377092.txt">Table of n, a(n) for n = 0..10000</a>
%H A377092 Rémy Sigrist, <a href="/A377092/a377092.png">Scatterplot of the first 50000 terms of the partial sums</a>
%H A377092 Rémy Sigrist, <a href="/A377092/a377092.gp.txt">PARI program</a>
%e A377092 The first terms are:
%e A377092    n   a(n)  |a(n)-a(n-1)|
%e A377092   --- ----- ---------------
%e A377092    0     0       N/A
%e A377092    1     1        1
%e A377092    2    -1        2
%e A377092    3     2        3
%e A377092    4     3        1
%e A377092    5    -2        5
%e A377092    6    -3        1
%e A377092    7    -4        1
%e A377092    8     4        8
%e A377092    9     5        1
%e A377092   10     6        1
%e A377092   11     7        1
%e A377092   12    -6       13
%e A377092   13    -5        1
%e A377092   14    -7        2
%e A377092 The first terms are a(0) = 0, a(1) = 1 and a(2) = -1, clearly the |smallest| unused number so far, which yields |a(2)-a(1)| = 2, a Fibonacci number. - _M. F. Hasler_, Feb 21 2025
%t A377092 A377092list[nmax_] := Module[{s, a, u = 1, fibQ},
%t A377092   fibQ[n_] := fibQ[n] = (IntegerQ[Sqrt[# + 4]] || IntegerQ[Sqrt[# - 4]]) & [5*n^2];
%t A377092   s[_] := False; s[0] = True;
%t A377092   NestList[(While[s[u] && s[-u], u++]; a = u; While[s[a] || !fibQ[Abs[# - a]], a = Boole[a < 0] - a]; s[a] = True; a) &, 0,nmax]];
%t A377092 A377092list[100] (* _Paolo Xausa_, Apr 19 2025 *)
%o A377092 (PARI) \\ See Links section.
%o A377092 (PARI) A377092_upto(N, U=[-1])={vector(N, n, if(n>1, for(k=U[1]+1,oo, A010056(k-N) && !setsearch(U, k) && [N=k, break]), N=0); U=setunion(U,[N]); while(#U>1&&U[1]+1==U[2],U=U[^1]); N)} \\ _M. F. Hasler_, Feb 21 2025
%o A377092 (Python)
%o A377092 def A377092(n):
%o A377092     if not getattr(A := A377092, 'N', 0):  A.N = 1; A.terms = [0]
%o A377092     while len(A.terms) <= n:
%o A377092         while (k := A.N) in A.terms: A.N = (k<0)-k
%o A377092         while not A010056(abs(k - A.terms[-1])) or k in A.terms: k = (k<0)-k
%o A377092         A.terms.append(k)
%o A377092     return A.terms[n] # _M. F. Hasler_, Feb 10 2025
%Y A377092 Cf. A000045, A377090, A377091, A380320 (first differences), A380321 (partial sums).
%Y A377092 Cf. A010056 (characteristic function of the Fibonacci numbers).
%K A377092 sign
%O A377092 0,4
%A A377092 _Rémy Sigrist_, Oct 16 2024