A027434 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.

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

Offset: 1

Author

Sam Speed (SPEEDS(AT)msci.memphis.edu)

Keywords

• nonn
• nice
• easy

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.

Crossrefs

Cf. A000267, A049068, A027709.

Cf. A082375, A350634.

Programs

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

Maple

A027434:=n->ceil(2*sqrt(n)); seq(A027434(n), n=1..100); # Wesley Ivan Hurt, Mar 01 2014

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 *)

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

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

a(n) = sqrt((A082375(n))^2 + 4*A350634(n+1)). - Frank M Jackson, Jan 21 2024

Extensions

More terms from Courtney Clipp (cclipp(AT)ashland.edu), Dec 08 2004