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A276862
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First differences of the Beatty sequence A003151 for 1 + sqrt(2).
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16
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2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2
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OFFSET
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1,1
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COMMENTS
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Theorem: If the initial term of A097509 is omitted, it is identical to the present sequence. For proof, see A097509. The argument may also imply that A082844 is also the same as these two sequences, apart from the initial terms. - Manjul Bhargava, Kiran Kedlaya, and Lenny Ng, Mar 02 2021. Postscript from the same authors, Sep 09 2021: We have proved that the present sequence, A276862 (indexed from 1) matches the characterization of {c_{i-1}} given by (8) of our "Solutions" page.
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LINKS
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FORMULA
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a(n) = floor((n+1)*r) - floor(n*r) = A003151(n+1)-A003151(n), where r = 1 + sqrt(2), n >= 1.
Fixed point of the morphism 2 -> 2,3; 3 -> 2,3,2. - John Keith, Apr 21 2021
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MATHEMATICA
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z = 500; r = 1+Sqrt[2]; b = Table[Floor[k*r], {k, 0, z}]; (* A003151 *)
Last@SubstitutionSystem[{2 -> {2, 3}, 3 -> {2, 3, 2}}, {2}, 5] (* John Keith, Apr 21 2021 *)
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PROG
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(PARI) vector(100, n, floor((n+1)*(1 + sqrt(2))) - floor(n*(1+sqrt(2)))) \\ G. C. Greubel, Aug 16 2018
(Magma) [Floor((n+1)*(1 + Sqrt(2))) - Floor(n*(1+Sqrt(2))): n in [1..100]]; // G. C. Greubel, Aug 16 2018
(Python)
from math import isqrt
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CROSSREFS
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The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A003151 as the parent: A003151, A001951, A001952, A003152, A006337, A080763, A082844 (conjectured), A097509, A159684, A188037, A245219 (conjectured), A276862. - N. J. A. Sloane, Mar 09 2021
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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