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

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, 38, 40, 41, 42, 44, 45, 46, 48, 50, 51, 53, 54, 55, 57, 58, 59, 61, 63, 64, 65, 67, 69, 70, 72, 73, 74, 76, 77, 78, 80, 82, 83, 84, 86, 88, 89, 91, 92, 93, 95, 97, 98, 100, 101, 102

Offset: 1

Author

Clark Kimberling, Jan 01 2008

Keywords

• nonn

Comments

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).

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.

Position of the n-th occurrence of either 1 or 3 in A105083(n-1) for n>=1. - Jeffrey Shallit, Mar 08 2025

Examples

			b(1) = a(a(1))+1 = a(1)+1 = 1+1 = 2;
b(2) = a(a(2))+2 = a(3)+2 = 4+2 = 6;
b(3) = a(a(3))+3 = a(4)+3 = 5+3 = 8;
b(4) = a(a(4))+4 = a(5)+4 = 7+4 = 11.
		

References

• 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.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Jeffrey Shallit, The Narayana Morphism and Related Words, arXiv:2503.01026 [math.CO], 2025.
• Eric Weisstein's World of Mathematics, Hofstadter H-Sequence.

Crossrefs

Cf. A005374, A020942, A035513, A105083, A136189, A136496.

Programs

Haskell

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a136495 n = (fromJust $ n `elemIndex` tail a005374_list) + 1
-- Reinhard Zumkeller, Dec 17 2011

Formula

A005374(a(n)) = n. - Reinhard Zumkeller, Dec 17 2011

a(n) = A005374(A005374(n-1)) + n. - Alan Michael Gómez Calderón, Jul 16 2025