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A136495 Solution of the complementary equation b(n)=a(a(n))+n.

%I A136495 #68 Jul 23 2025 00:12:05
%S A136495 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 A136495 38,40,41,42,44,45,46,48,50,51,53,54,55,57,58,59,61,63,64,65,67,69,70,
%U A136495 72,73,74,76,77,78,80,82,83,84,86,88,89,91,92,93,95,97,98,100,101,102
%N A136495 Solution of the complementary equation b(n)=a(a(n))+n.
%C A136495 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 A136495 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.
%C A136495 Position of the n-th occurrence of either 1 or 3 in A105083(n-1) for n>=1. - _Jeffrey Shallit_, Mar 08 2025
%D A136495 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 A136495 Reinhard Zumkeller, <a href="/A136495/b136495.txt">Table of n, a(n) for n = 1..10000</a>
%H A136495 Jeffrey Shallit, <a href="https://arxiv.org/abs/2503.01026">The Narayana Morphism and Related Words</a>, arXiv:2503.01026 [math.CO], 2025.
%H A136495 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HofstadterH-Sequence.html">Hofstadter H-Sequence.</a>
%F A136495 A005374(a(n)) = n. - _Reinhard Zumkeller_, Dec 17 2011
%F A136495 a(n) = A005374(A005374(n-1)) + n. - _Alan Michael Gómez Calderón_, Jul 16 2025
%e A136495 b(1) = a(a(1))+1 = a(1)+1 = 1+1 = 2;
%e A136495 b(2) = a(a(2))+2 = a(3)+2 = 4+2 = 6;
%e A136495 b(3) = a(a(3))+3 = a(4)+3 = 5+3 = 8;
%e A136495 b(4) = a(a(4))+4 = a(5)+4 = 7+4 = 11.
%o A136495 (Haskell)
%o A136495 import Data.List (elemIndex)
%o A136495 import Data.Maybe (fromJust)
%o A136495 a136495 n = (fromJust $ n `elemIndex` tail a005374_list) + 1
%o A136495 -- _Reinhard Zumkeller_, Dec 17 2011
%Y A136495 Cf. A005374, A020942, A035513, A105083, A136189, A136496.
%K A136495 nonn
%O A136495 1,2
%A A136495 _Clark Kimberling_, Jan 01 2008