A006338 An "eta-sequence": floor((n+1)*sqrt(2) + 1/2) - floor(n*sqrt(2) + 1/2). (Formerly M0087)

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

1,1

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

Formula

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

Prog

(Haskell)
a006338 n = a006338_list !! (n-1)
a006338_list = tail a214848_list
-- Reinhard Zumkeller, Mar 03 2014
(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
(Magma) [Floor((n+1)*Sqrt(2)+1/2) - Floor(n*Sqrt(2)+1/2): n in [1..30]]; // G. C. Greubel, Nov 18 2017

Crossrefs

Cf. A006337, A022846, A214848.

Sequence in context: A214856 A006337 A214848 * A020903 A133083 A083921

Adjacent sequences: A006335 A006336 A006337 * A006339 A006340 A006341

Keyword

nonn,easy,nice

Author

D. R. Hofstadter, Jul 15 1977

Extensions

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

Status

approved