A006338 An "eta-sequence": floor((n+1)*sqrt(2) + 1/2) - floor(n*sqrt(2) + 1/2).

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, 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, 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

Offset: 1

Author

D. R. Hofstadter, Jul 15 1977

Keywords

• nonn
• easy
• nice

Comments

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

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.

a(n) = A214848(n) = A022846(n+1) - A022846(n). - Reinhard Zumkeller, Mar 03 2014

References

• Douglas Hofstadter, "Fluid Concepts and Creative Analogies", Chapter 1: "To seek whence cometh a sequence".
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• D. R. Hofstadter, Eta-Lore [Cached copy, with permission]
• D. R. Hofstadter, Pi-Mu Sequences [Cached copy, with permission]
• D. R. Hofstadter and N. J. A. Sloane, Correspondence, 1977 and 1991

Crossrefs

Cf. A006337, A022846, A214848.

Programs

Haskell

a006338 n = a006338_list !! (n-1)
a006338_list = tail a214848_list
-- Reinhard Zumkeller, Mar 03 2014

Magma

[Floor((n+1)*Sqrt(2)+1/2) - Floor(n*Sqrt(2)+1/2): n in [1..30]]; // G. C. Greubel, Nov 18 2017

Mathematica

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 *)
Differences[Table[Floor[n Sqrt[2]+1/2],{n,120}]] (* Harvey P. Dale, Dec 10 2021 *)

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

Formula

a(n) = floor((n+1)*sqrt(2) + 1/2) - floor(n*sqrt(2) + 1/2). - G. C. Greubel, Nov 18 2017

Extensions

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