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A136495 Solution of the complementary equation b(n)=a(a(n))+n. 4
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 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A258932 A183213 A183172 * A184419 A285970 A189665
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jan 01 2008
STATUS
approved

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Last modified April 12 17:42 EDT 2024. Contains 371635 sequences. (Running on oeis4.)