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A305820 Filter sequence for a(Fibonacci numbers > 1) = constant sequences.

%I A305820 #7 Jun 16 2018 18:30:42
%S A305820 0,1,2,2,3,2,4,5,2,6,7,8,9,2,10,11,12,13,14,15,16,2,17,18,19,20,21,22,
%T A305820 23,24,25,26,27,28,2,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,
%U A305820 45,46,47,48,2,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,2,82,83,84,85,86
%N A305820 Filter sequence for a(Fibonacci numbers > 1) = constant sequences.
%C A305820 For all i, j:
%C A305820   a(i) = a(j) => A003603(i) = A003603(j) => A304101(i) = A304101(j) => A007895(i) = A007895(j).
%F A305820 a(0)= 0, a(1) = 1, for n > 1, a(n) = 2 if n is a Fibonacci number > 1, otherwise a(n) = 2+n-A072649(n) = running count from 3 onward for non-Fibonacci numbers.
%o A305820 (PARI)
%o A305820 A010056(n) = { my(k=n^2); k+=(k+1)<<2; (issquare(k) || (n>0 && issquare(k-8))) }; \\ From A010056
%o A305820 A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
%o A305820 A305820(n) = if(n<=1, n, if(1==A010056(n),2,2+n-A072649(n)));
%Y A305820 Cf. A000045, A003603, A007895, A010056, A072649, A304101.
%K A305820 nonn
%O A305820 0,3
%A A305820 _Antti Karttunen_, Jun 16 2018