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A136495
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Solution of the complementary equation b(n)=a(a(n))+n.
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4
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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
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OFFSET
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1,2
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COMMENTS
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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).
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REFERENCES
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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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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.
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PROG
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(Haskell)
import Data.List (elemIndex)
import Data.Maybe (fromJust)
a136495 n = (fromJust $ n `elemIndex` tail a005374_list) + 1
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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