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A027434
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a(1) = 2; then defined by property that a(n) = smallest number >= a(n-1) such that successive runs have lengths 1,1,2,2,3,3,4,4.
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14
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2, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18
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listen;
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OFFSET
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1,1
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COMMENTS
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Also the sequence of first skipped terms for Beatty sequences in the family alpha = 1+sqrt(n)-sqrt(n-1). - Alisa Ediger, Jul 20 2016
Optimal cost for one-dimensional Racetrack over a distance n. - Jason Schoeters, Aug 18 2021
If b > 0 and c > 0 are the integer coefficients of a monic quadratic x^2 + b*x + c, it has integer roots if its discriminant d^2 = b^2 - 4c is a perfect square. This sequence is the values of b for increasing b sorted by b then c. The first pair of (b, c) = (2, 1) and has d = A082375(0) = 0. The n-th pair of (b, c) = (a(n), A350634(n)) and has d = A082375(n-1). - Frank M Jackson, Jan 21 2024
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REFERENCES
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Sam Speed, An integer sequence (preprint).
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LINKS
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FORMULA
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a(n) = 1 + floor( sqrt(4*n-3) ) = 1+A000267(n-1).
a(n) = floor(1+sqrt(n)+sqrt(n-1)). - Alisa Ediger, Jul 20 2016
G.f.: x*(1 + x^(-1/4)*theta_2(x) + theta_3(x))/(2*(1 - x)), where theta_k(x) is the Jacobi theta function. - Ilya Gutkovskiy, Jul 20 2016
a(n) = 1 + floor(sqrt(4*n-1)). - Chai Wah Wu, Jul 27 2022
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MAPLE
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MATHEMATICA
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Sort[Flatten[Table[#, {#[[1]]/2}]]]&/@Partition[Range[2, 20], 2]//Flatten (* Harvey P. Dale, Sep 05 2019 *)
lst = {}; Do[If[IntegerQ[d=Sqrt[b^2-4 c]], AppendTo[lst, b]], {b, 1, 20}, {c, 1, b^2/4}]; lst (* Frank M Jackson, Jan 21 2024 *)
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PROG
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(Haskell)
a027434 = (+ 1) . a000196 . (subtract 3) . (* 4)
a027434_list = 2 : concat (map (\x -> replicate (x `div` 2) x) [3..])
(Python)
from math import isqrt
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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Sam Speed (SPEEDS(AT)msci.memphis.edu)
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EXTENSIONS
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More terms from Courtney Clipp (cclipp(AT)ashland.edu), Dec 08 2004
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STATUS
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approved
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