OFFSET
1,1
COMMENTS
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
REFERENCES
Sam Speed, An integer sequence (preprint).
LINKS
William A. Tedeschi, Table of n, a(n) for n = 1..10000
A. Casteigts, M. Raffinot and J. Schoeters, VectorTSP: A Traveling Salesperson Problem with Racetrack-like acceleration constraints, Lemma 7, arXiv:2006.03666 [cs.DS], 2020-2021.
FORMULA
a(n) = 1 + floor( sqrt(4*n-3) ) = 1+A000267(n-1).
a(n) = A049068(n) - n.
a(n) = A027709(n)/2. - Tanya Khovanova, Mar 04 2008
a(n) = ceiling(2*sqrt(n)). [Mircea Merca, Feb 07 2012]
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
MAPLE
MATHEMATICA
Table[Ceiling[2*Sqrt[n]], {n, 100}] (* Wesley Ivan Hurt, Mar 01 2014 *)
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 *)
PROG
(Haskell)
a027434 = (+ 1) . a000196 . (subtract 3) . (* 4)
a027434_list = 2 : concat (map (\x -> replicate (x `div` 2) x) [3..])
-- Reinhard Zumkeller, Mar 23 2013, Nov 22 2011
(PARI) a(n)=sqrtint(4*n-3)+1 \\ Charles R Greathouse IV, Feb 07 2012
(Python)
from math import isqrt
def A027434(n): return 1+isqrt((n<<2)-1) # Chai Wah Wu, Jul 27 2022
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
Sam Speed (SPEEDS(AT)msci.memphis.edu)
EXTENSIONS
More terms from Courtney Clipp (cclipp(AT)ashland.edu), Dec 08 2004
STATUS
approved