A006338 - OEIS

%I M0087 #33 Sep 08 2022 08:44:34

%S 2,1,2,1,1,2,1,2,1,2,1,1,2,1,2,1,1,2,1,2,1,2,1,1,2,1,2,1,1,2,1,2,1,1,

%T 2,1,2,1,2,1,1,2,1,2,1,1,2,1,2,1,2,1,1,2,1,2,1,1,2,1,2,1,2,1,1,2,1,2,

%U 1,1,2,1,2,1,1,2,1,2,1,2,1,1,2,1,2,1,1,2,1,2,1,2,1,1,2,1,2,1,1,2,1,2,1,1,2

%N An "eta-sequence": floor((n+1)*sqrt(2) + 1/2) - floor(n*sqrt(2) + 1/2).

%C Equals its own "second derivative" (cf. A006337).

%C Presumably this is the same as the following sequence from Hofstadter's book: the number of triangular numbers between each successive pair of squares. More precisely, a(n) is the number of triangular numbers T such that n^2 <= T < (n+1)^2. E.g., a(3) = 2 because 3^2 <= T < 4^2 permits T(4) = 10 and T(5) = 15 and no other triangular number. - _Hugo van der Sanden_, May 03 2005.

%C a(n) = A214848(n) = A022846(n+1) - A022846(n). - _Reinhard Zumkeller_, Mar 03 2014

%D Douglas Hofstadter, "Fluid Concepts and Creative Analogies", Chapter 1: "To seek whence cometh a sequence".

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Reinhard Zumkeller, <a href="/A006338/b006338.txt">Table of n, a(n) for n = 1..10000</a>

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

%H D. R. Hofstadter, <a href="/A006336/a006336_2.pdf">Pi-Mu Sequences</a> [Cached copy, with permission]

%H D. R. Hofstadter and N. J. A. Sloane, <a href="/A006336/a006336.pdf">Correspondence, 1977 and 1991</a>

%F a(n) = floor((n+1)*sqrt(2) + 1/2) - floor(n*sqrt(2) + 1/2). - _G. C. Greubel_, Nov 18 2017

%t a[n_] := Floor[(n+1)*Sqrt[2]+1/2] - Floor[n*Sqrt[2]+1/2]; Table[a[n], {n, 1, 105}] (* _Jean-François Alcover_, Nov 24 2015 *)

%t Differences[Table[Floor[n Sqrt[2]+1/2],{n,120}]] (* _Harvey P. Dale_, Dec 10 2021 *)

%o (Haskell)

%o a006338 n = a006338_list !! (n-1)

%o a006338_list = tail a214848_list

%o -- _Reinhard Zumkeller_, Mar 03 2014

%o (PARI) for(n=1,30, print1(floor((n+1)*sqrt(2) + 1/2) - floor(n*sqrt(2) + 1/2), ", ")) \\ _G. C. Greubel_, Nov 18 2017

%o (Magma) [Floor((n+1)*Sqrt(2)+1/2) - Floor(n*Sqrt(2)+1/2): n in [1..30]]; // _G. C. Greubel_, Nov 18 2017

%Y Cf. A006337, A022846, A214848.

%K nonn,easy,nice

%O 1,1

%A D. R. Hofstadter, Jul 15 1977

%E More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 28 2003