A063957 Numbers not of the form round(m*sqrt(2)) for any integer m, i.e., complement of A022846.

2, 5, 9, 12, 15, 19, 22, 26, 29, 32, 36, 39, 43, 46, 50, 53, 56, 60, 63, 67, 70, 73, 77, 80, 84, 87, 90, 94, 97, 101, 104, 108, 111, 114, 118, 121, 125, 128, 131, 135, 138, 142, 145, 149, 152, 155, 159, 162, 166, 169, 172, 176, 179, 183, 186, 189, 193, 196, 200

Offset: 1

Author

Henry Bottomley, Sep 04 2001

Keywords

• nonn

Comments

Consider natural numbers A000027 as a triangle 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, etc., then the a(n) indicate rows without a square.

Similar to Beatty sequences: where a pair of complementary Beatty sequences are floor(n*c) and floor(n*c/(c-1)) for c an irrational constant > 1, these pairs of complementary sequences are in general round(n*c) and round((n-1/2)*c/(c-1)) for c an irrational constant > 1.

This sequence is an inhomogeneous Beatty sequence s(alpha,rho) with slope alpha = 2 + sqrt(2), and intercept rho = -1/2 - sqrt(2)/2. - Michel Dekking, Sep 15 2022

Let D := 3,4,3,3,4,3,4,3,3,4,3,4,3,4,3,3,4,3,... be the sequence of first differences of (a(n)). It follows from Yasutomi's criterion that D is NOT the fixed point of a morphism. - Michel Dekking, Sep 20 2022

Examples

			round(m*sqrt(2)) starts 1,3,4,6,7,8,10,11,13,... so this sequence must start 2,5,9,12,...
		

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000
• A. S. Fraenkel, The bracket function and complementary sets of integers, Canadian J. of Math. 21 (1969) 6-27. (Theorem XI)
• Clark Kimberling, Beatty sequences and trigonometric functions, Integers 16 (2016), #A15.
• S.-I. Yasutomi, On Sturmian sequences which are invariant under some substitutions, on ResearchGate.

Crossrefs

Cf. A001951, A001952, A007064, A022846.

Programs

PARI

{ f=2 + sqrt(2); t=f/2; for (n=1, 1000, write("b063957.txt", n, " ", round(n*f - t)) ) } \\ Harry J. Smith, Sep 03 2009

Python

from math import isqrt
def A063957(n): return (a:=(n<<1)-1)+(m:=isqrt(k:=a**2<<1)>>1)+int(((m<<1)+1)**2Chai Wah Wu, Feb 11 2025

Formula

a(n) = round((n - 1/2)*(2 + sqrt(2))) = round(n*3.4142...-1.7071...).