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A214848
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First difference of A022846.
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5
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1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1
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OFFSET
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0,2
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COMMENTS
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Number of triangular numbers in interval [n^2, (n+1)^2).
(a(n)) is an inhomogeneous Sturmian sequence s(alpha, rho) with slope alpha = sqrt(2) and intercept 1/2, since A022846(n) = floor(n*sqrt(2) + 1/2).
(a(n)) is the fixed point of the morphism 1->12121, 2->1212121.
This is proved by writing the 0-1 version psi: 0->01010, 1->0101010 of this morphism as a composition
psi = psi_1 psi_3 psi_1 psi_4,
where the psi_i are the three elementary Sturmian morphisms
psi_1: 0->01, 1->0, psi_3: 0->0, 1->01, psi_4: 0->0, 1->10.
By Lemma 2.2.18 in Lothaire it then follows that the 0-1 word (a(n)-1) = A214848 is fixed by the morphism psi (note that in Lothaire psi_1 is phi, psi_3 is G, and psi_4 is G^~). (End)
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REFERENCES
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S.-I. Yasutomi, On Sturmian sequences which are invariant under some substitutions, in Number theory and its applications (Kyoto, 1997), pp. 347-373, Kluwer Acad. Publ., Dordrecht, 1999.
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LINKS
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FORMULA
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EXAMPLE
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28 is in [25, 36), a(5) = 1.
36 and 45 are in [36, 49), a(6) = 2.
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MATHEMATICA
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Differences[Round[Sqrt[2]Range[0, 100]]] (* Harvey P. Dale, Jun 14 2020 *)
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PROG
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(Haskell)
a214848 n = a214848_list !! n
a214848_list = zipWith (-) (tail a022846_list) a022846_list
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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