A197878 a(n) = floor(2*(1 + sqrt(2))*n).

4, 9, 14, 19, 24, 28, 33, 38, 43, 48, 53, 57, 62, 67, 72, 77, 82, 86, 91, 96, 101, 106, 111, 115, 120, 125, 130, 135, 140, 144, 149, 154, 159, 164, 168, 173, 178, 183, 188, 193, 197, 202, 207, 212, 217, 222, 226, 231, 236, 241, 246, 251, 255, 260, 265, 270

Offset: 1

Author

Zak Seidov, Oct 18 2011

Keywords

• nonn
• easy

Comments

First differences are 4 and 5. Also, there is no immediate pattern in parity of a(n).

Are similar sequences well defined (in terms of rounding problems)? See also A086843, A086844, A196468.

Answer: I would not call the sequences A086843, A086844, A196468 'similar' to (a(n)). The first differences d =5,5,5,5,4,5,5,5,5,4,... are a Sturmian sequence (d(n)) with slope alpha = 2 + sqrt(8) and intercept 0. We give d offset 0 by setting d(0):=4. By Hofstadter's Fundamental Theorem of eta-sequences, the chunks 45555 and 455555 occur following a Sturmian sequence with density beta = (sqrt(8) - 2)/(3 - sqrt(8)). Since beta = 2 + sqrt(8) = alpha, the sequence (d(n)) is fixed point of the substitution 4->45555, 5->455555. See A197879 for a complete description of the parity pattern of (a(n)). - Michel Dekking, Jan 24 2017

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• J. N. Cooper and A. W. N. Riasanovsky, On the Reciprocal of the Binary Generating Function for the Sum of Divisors, 2012; J. Int. Seq. 16 (2013) #13.1.8
• D. R. Hofstadter, Eta-Lore [Cached copy, with permission]
• N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

Crossrefs

Cf. A001030. - Michel Dekking, Jan 24 2017

Cf. A086843, A086844, A196468.

A bisection of A003151.

Programs

Magma

[Floor(2*(1 + Sqrt(2))*n): n in [1..100]]; // G. C. Greubel, Aug 18 2018

Mathematica

Table[Floor[((2+Sqrt[8]))*n], {n,100}]

PARI

a(n)=2*n+sqrtint(8*n^2) \\ Charles R Greathouse IV, Oct 25 2011

Formula

a(n) = A003151(2n). - R. J. Mathar, Oct 20 2011