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A345290 a(n) is obtained by replacing 2^k in binary expansion of n with Fibonacci(-k-2).

%I A345290 #13 Jun 17 2021 09:11:56
%S A345290 0,-1,2,1,-3,-4,-1,-2,5,4,7,6,2,1,4,3,-8,-9,-6,-7,-11,-12,-9,-10,-3,
%T A345290 -4,-1,-2,-6,-7,-4,-5,13,12,15,14,10,9,12,11,18,17,20,19,15,14,17,16,
%U A345290 5,4,7,6,2,1,4,3,10,9,12,11,7,6,9,8,-21,-22,-19,-20,-24
%N A345290 a(n) is obtained by replacing 2^k in binary expansion of n with Fibonacci(-k-2).
%C A345290 This sequence is a variant of A022290; here we consider Fibonacci numbers with negative indices (A039834), there Fibonacci numbers with positive indices (A000045).
%C A345290 After the initial 0, the sequence alternates runs of positive terms and runs of negative terms, the k-th run having 2^(k-1) terms.
%H A345290 Rémy Sigrist, <a href="/A345290/b345290.txt">Table of n, a(n) for n = 0..8191</a>
%F A345290 a(n) = A022290(A063695(n)) - A022290(A063694(n)).
%F A345290 a(n) = A022290(n) iff n belongs to A062880.
%F A345290 a(n) = -A022290(n) iff n belongs to A000695.
%F A345290 a(n) = 0 iff n = 0.
%F A345290 a(n) = 1 iff n belongs to A072197.
%F A345290 a(n) = 2 iff n belongs to A080675.
%F A345290 a(n) = -1 iff n belongs to A020989.
%F A345290 a(n) = -2 iff n belongs to A136412.
%e A345290 For n = 3:
%e A345290 - 3 = 2^1 + 2^0,
%e A345290 - so a(3) = A039834(2+1) + A039834(2+0) = 2 - 1 = 1.
%o A345290 (PARI) a(n) = { my (v=0, e); while (n, n-=2^e=valuation(n, 2); v+=fibonacci(-2-e)); v }
%Y A345290 Cf. A000045, A000695, A020989, A022290, A039834, A062880, A063694, A063695, A072197, A080675, A136412, A345291, A345292.
%K A345290 sign,base
%O A345290 0,3
%A A345290 _Rémy Sigrist_, Jun 13 2021