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Search: id:A116576
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| A116576 |
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Number of distinct squares D(n) in the n-th iterate of the tribonacci morphism (a -> ab, b -> ac, c -> a) on the letter a. |
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+0
1
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| 2, 7, 17, 35, 69, 132, 248, 462, 856, 1581, 2915, 5369, 9883, 18186, 33458, 61548, 113214, 208243, 383029, 704511, 1295809, 2383376, 4383724, 8062938, 14830068, 27276761, 50169799, 92276661, 169723255, 312169750, 574169702, 1056062744
(list; graph; listen)
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OFFSET
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4,1
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COMMENT
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See the cited reference (A. Glen) for similar sequences associated to the k-bonacci morphism and particular episturmian (infinite) words.
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REFERENCES
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Amy Glen, On Sturmian and episturmian words, and related topics, Ph.D. Thesis, The Univeristy of Adelaide (Australia), April 2006.
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FORMULA
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For n >= 6, D(n) = d(n-5) + d(n-6) + 1 + Sum (d(m) + 1), m = 0 ... (n-3), where d(m) = (T(m+1) + T(m-1) - 1)/2 - 1 and the T(m) are the tribonacci numbers: (T(0), T(1), T(2), ...) = (1, 2, 4, 7, 13, ...).
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EXAMPLE
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D(4) = 2 because the 4-th iterate of the tribonacci morphism on a is abacabaabacab, which contains the two squares aa and abaaba.
D(5) = 7 because the 5-th iterate of the tribonacci morphism on a is abacabaabacababacabaabac, which contains the square of each of the following seven strings: a, ab, ba, aba, abacab, bacaba, abacaba.
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PROGRAM
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(MatLab) T(1) = 1; T(2) = 2; T(3) = 4; for n = 4 : 100 T(n) = T(n-1) + T(n-2) + T(n-3); end; d(1) = 0; for n = 2 : 50 d(n) = (T(n+1) + T(n-1) - 1)/2 - 1; end; D(4) = 2; D(5) = 7; for n = 6 : 50 D(n) = d(n-4) + d(n-5) + 1 + sum(d(1:(n-2))) + (n - 2); end; disp(num2str(D));
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CROSSREFS
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Adjacent sequences: A116573 A116574 A116575 this_sequence A116577 A116578 A116579
Sequence in context: A014148 A070070 A033937 this_sequence A086717 A023657 A045380
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KEYWORD
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nonn,uned
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AUTHOR
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Amy Glen (amy.glen(AT)gmail.com), Apr 07 2006
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