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%I #16 Mar 13 2021 10:55:59
%S 1,3,4,5,7,9,10,12,13,14,16,17,18,20,22,23,24,26,28,29,31,32,33,35,37,
%T 38,40,41,42,44,45,46,48,50,51,53,54,55,57,58,59,61,63,64,65,67,69,70,
%U 72,73,74,76,77,78,80,82,83,84,86,88,89,91,92,93,95,97,98,100,101,102
%N Solution of the complementary equation b(n)=a(a(n))+n.
%C b = 1 + (column 1 of Z) = 1 + A020942. The pair (a,b) also satisfy the following complementary equations: b(n)=a(a(a(n)))+1; a(b(n))=a(n)+b(n); b(a(n))=a(n)+b(n)-1; (and others).
%C A005374(a(n)) = n. [_Reinhard Zumkeller_, Dec 17 2011]
%D Clark Kimberling and Peter J. C. Moses, Complementary equations and Zeckendorf arrays, in Applications of Fibonacci Numbers, vol.10, Proceedings of the Thirteenth International Conference on Fibonacci Numbers and Their Applications, William Webb, editor, Congressus Numerantium, Winnipeg, Manitoba 201 (2010) 161-178.
%H Reinhard Zumkeller, <a href="/A136495/b136495.txt">Table of n, a(n) for n = 1..10000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HofstadterH-Sequence.html">Hofstadter H-Sequence.</a>
%F Let Z = (3rd order Zeckendorf array) = A136189. Then a = ordered union of columns 1,3,4,6,7,9,10,12,13,... of Z, b = ordered union of columns 2,5,8,11,14,... of Z.
%e b(1) = a(a(1))+1 = a(1)+1 = 1+1 = 2;
%e b(2) = a(a(2))+2 = a(3)+2 = 4+2 = 6;
%e b(3) = a(a(3))+3 = a(4)+3 = 5+3 = 8;
%e b(4) = a(a(4))+4 = a(5)+4 = 7+4 = 11.
%o (Haskell)
%o import Data.List (elemIndex)
%o import Data.Maybe (fromJust)
%o a136495 n = (fromJust $ n `elemIndex` tail a005374_list) + 1
%o -- _Reinhard Zumkeller_, Dec 17 2011
%Y Cf. A020942, A035513, A136189, A136496.
%K nonn
%O 1,2
%A _Clark Kimberling_, Jan 01 2008
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