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A214974 Numbers k for which A116543(k) = A007895(k); i.e., the Lucas and Zeckendorf representations of k have the same length.

%I A214974 #5 Dec 04 2016 19:46:30
%S A214974 1,2,3,6,9,10,14,15,17,22,27,28,36,38,41,43,44,46,52,58,59,61,62,66,
%T A214974 69,74,75,81,84,94,95,96,98,107,112,114,117,119,120,122,128,131,136,
%U A214974 139,148,152,153,154,155,159,161,164,173,175,176,181,182,184,185
%N A214974 Numbers k for which A116543(k) = A007895(k); i.e., the Lucas and Zeckendorf representations of k have the same length.
%H A214974 Clark Kimberling, <a href="/A214974/b214974.txt">Table of n, a(n) for n = 1..8000</a>
%e A214974 k...Lucas.....Zeckendorf....counter
%e A214974 1...1.........1.............a(1)= 1
%e A214974 2...2.........2.............a(2)= 2
%e A214974 3...3.........3.............a(3)= 3
%e A214974 4...4.........3+1
%e A214974 5...4+1.......5
%e A214974 6...4+2.......5+1...........a(4)= 6
%e A214974 7...7.........5+2
%e A214974 8...7+1.......8
%e A214974 9...7+2.......8+1...........a(5)= 9
%t A214974 u = Reverse[Sort[Table[LucasL[n - 1], {n, 1, 50}]]];
%t A214974 u1 = Map[Length[Select[Reap[FoldList[(Sow[Quotient[#1, #2]]; Mod[#1, #2]) &, #, u]][[2,1]], # > 0 &]] &, Range[1000]];
%t A214974 v = Reverse[Table[Fibonacci[n + 1], {n, 1, 50}]];
%t A214974 v1 = Map[Length[Select[Reap[FoldList[(Sow[Quotient[#1, #2]]; Mod[#1, #2]) &, #, v]][[2,1]], # > 0 &]] &, Range[1000]];
%t A214974 s[n_] := If[u1[[n]] == v1[[n]], 1, 0];
%t A214974 s1 = Table[s[n], {n, 1, 200}];
%t A214974 f1 = Flatten[Position[s1, 1]] (* A214974 *)
%t A214974 s[n_] := If[u1[[n]] < v1[[n]], 1, 0];
%t A214974 s2 = Table[s[n], {n, 1, 200}];
%t A214974 f2 = Flatten[Position[s2, 1]] (* A214975 *)
%t A214974 s[n_] := If[u1[[n]] > v1[[n]], 1, 0];
%t A214974 s3 = Table[s[n], {n, 1, 200}];
%t A214974 f3 = Flatten[Position[s3, 1]] (* A214976 *)
%t A214974 (* _Peter J. C. Moses_ *)
%Y A214974 Cf. A116543, A007895, A214975, A214976, A214973.
%K A214974 nonn
%O A214974 1,2
%A A214974 _Clark Kimberling_, Oct 20 2012