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

1,2

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).

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

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

Eric Weisstein's World of Mathematics, Hofstadter H-Sequence.

Formula

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.

Example

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.

Prog

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

Crossrefs

Cf. A020942, A035513, A136189, A136496.

Sequence in context: A258932 A183213 A183172 * A184419 A285970 A189665

Adjacent sequences: A136492 A136493 A136494 * A136496 A136497 A136498

Keyword

nonn

Author

Clark Kimberling, Jan 01 2008

Status

approved