%I #41 Sep 08 2022 08:45:59

%S 4,9,14,19,24,28,33,38,43,48,53,57,62,67,72,77,82,86,91,96,101,106,

%T 111,115,120,125,130,135,140,144,149,154,159,164,168,173,178,183,188,

%U 193,197,202,207,212,217,222,226,231,236,241,246,251,255,260,265,270

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

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

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

%C 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

%H Vincenzo Librandi, <a href="/A197878/b197878.txt">Table of n, a(n) for n = 1..10000</a>

%H J. N. Cooper and A. W. N. Riasanovsky, <a href="http://www.math.sc.edu/~cooper/Sigma.pdf">On the Reciprocal of the Binary Generating Function for the Sum of Divisors</a>, 2012; <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Cooper/cooper3.html">J. Int. Seq. 16 (2013) #13.1.8</a>

%H D. R. Hofstadter, <a href="/A006336/a006336_1.pdf">Eta-Lore</a> [Cached copy, with permission]

%H N. J. A. Sloane, <a href="/A115004/a115004.txt">Families of Essentially Identical Sequences</a>, Mar 24 2021 (Includes this sequence)

%F a(n) = A003151(2n). - _R. J. Mathar_, Oct 20 2011

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

%o (PARI) a(n)=2*n+sqrtint(8*n^2) \\ _Charles R Greathouse IV_, Oct 25 2011

%o (Magma) [Floor(2*(1 + Sqrt(2))*n): n in [1..100]]; // _G. C. Greubel_, Aug 18 2018

%Y Cf. A001030. - _Michel Dekking_, Jan 24 2017

%Y Cf. A086843, A086844, A196468.

%Y A bisection of A003151.

%K nonn,easy

%O 1,1

%A _Zak Seidov_, Oct 18 2011