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A330037
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The sum of digits function modulo 2 of the natural numbers in base phi.
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2
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0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0
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OFFSET
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0
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COMMENTS
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This sequence is a morphic sequence, i.e., a letter-to-letter image of a fixed point of a morphism.
Let the morphism tau on the alphabet A:={1,2,...,8} be defined by
tau(1) = 12, tau(2) = 312, tau(3) = 47, tau(4) = 8312,
tau(5) = 56, tau(6) = 756, tau(7) = 83, tau(8) = 4756.
Let lambda: A* -> {0,1} be the letter-to-letter morphism given by
lambda(1) = lambda(3) = lambda(6) = lambda(8) = 0;
lambda(2) = lambda(4) = lambda(5) = lambda(7) = 1.
Then a(n) = lambda(x(n)), where x(0)x(1)... = 123124712... is the fixed point of tau starting with 1. For a proof, see "The sum of digits function of the base phi expansion of the natural numbers".
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LINKS
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FORMULA
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EXAMPLE
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In base phi: 2 = 10.01, so a(2)=0; 3 = 100.01 so a(3)=0.
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CROSSREFS
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See A130600 for the integers written in base phi, with the "decimal point" omitted. See A105424 for the part of n in base phi left of the decimal point.
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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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