A196199 Count up from -n to n for n = 0, 1, 2, ... .
0, -1, 0, 1, -2, -1, 0, 1, 2, -3, -2, -1, 0, 1, 2, 3, -4, -3, -2, -1, 0, 1, 2, 3, 4, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8
Offset: 0
Examples
Table starts:
0,
-1, 0, 1,
-2, -1, 0, 1, 2,
-3, -2, -1, 0, 1, 2, 3,
...
The sequence of fractions A196199/A004737 = 0/1, -1/1, 0/2, 1/1, -2/1, -1/2, 0/3, 1/2, 2/1, -3/1, -2/2, -1/3, 0/4, 1/3, 2/2, 3/1, -4/4. -3/2, ... contains every rational number (infinitely often) [Laczkovich]. - _N. J. A. Sloane_, Oct 09 2013
References
- Miklós Laczkovich, Conjecture and Proof, TypoTex, Budapest, 1998. See Chapter 10.
Links
- Reinhard Zumkeller, Rows n=0..100 of triangle, flattened
- Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Crossrefs
Programs
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Haskell
a196199 n k = a196199_row n !! k a196199_tabf = map a196199_row [0..] a196199_row n = [-n..n] b196199 = bFile' "A196199" (concat $ take 101 a196199_tabf) 0 -- Reinhard Zumkeller, Sep 30 2011
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Maple
seq(seq(j-k-k^2, j=k^2 .. (k+1)^2-1), k = 0 .. 10); # Robert Israel, Jan 05 2015 # Alternatively, as a table with rows -n<=k<=n (compare A257564): r := n -> (n-(n mod 2))/2: T := (n, k) -> r(n+k) - r(n-k): seq(print(seq(T(n, k), k=-n..n)), n=0..6); # Peter Luschny, May 28 2015
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Mathematica
Table[Range[-n, n], {n, 0, 9}] // Flatten (* or *) a[n_] := With[{t = Floor[Sqrt[n]]}, n - t (t + 1)]; Table[a[n], {n, 0, 99}] (* Jean-François Alcover, Jul 13 2018, after Boris Putievskiy *) -
PARI
r=[];for(k=0,8,r=concat(r,vector(2*k+1,j,j-k-1)));r
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Python
from math import isqrt def A196199(n): return n-(t:=isqrt(n))*(t+1) # Chai Wah Wu, Aug 04 2022
Formula
a(n) = n - t*t - t - 1, where t = floor(sqrt(n-1)). - Boris Putievskiy, Jan 28 2013
G.f.: x/(x-1)^2 + 1/(x-1)*sum(k >= 1, 2*k*x^(k^2)). The series is related to Jacobi theta functions. - Robert Israel, Jan 05 2015
Comments