%I #9 Mar 13 2021 10:54:04
%S 2,6,8,11,15,19,21,25,27,30,34,36,39,43,47,49,52,56,60,62,66,68,71,75,
%T 79,81,85,87,90,94,96,99,103,107,109,113,115,118,122,124,127,131,135,
%U 137,140,144,148,150,154,156,159,163,165,168,172,176,178,181,185,189
%N Solution of the complementary equation b(n)=a(a(n))+n; this is sequence b; sequence a is A136495.
%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).
%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.
%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.
%Y Cf. A020942, A035513, A136189, A136495.
%K nonn
%O 1,1
%A _Clark Kimberling_, Jan 01 2008
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