A381232 -id:A381232 - cp's OEIS frontend

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

Franklin T. Adams-Watters, Sep 29 2011

This sequence contains every integer infinitely often, hence all integer sequences are subsequences.
This is a fractal sequence.
Indeed, if all terms (a(n),a(n+1)) such that a(n+1) does NOT equal a(n)+1 (<=> a(n+1) < a(n)) are deleted, the same sequence is recovered again. See A253580 for an "opposite" yet similar "fractal tree" construction. - M. F. Hasler, Jan 04 2015

			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

Miklós Laczkovich, Conjecture and Proof, TypoTex, Budapest, 1998. See Chapter 10.

Reinhard Zumkeller, Rows n=0..100 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Cf. absolute values A053615, A002262, A002260, row lengths A005408, row sums A000004, A071797.

Cf. A004737, A381232, A381233.

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

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

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 *)

r=[];for(k=0,8,r=concat(r,vector(2*k+1,j,j-k-1)));r

from math import isqrt
def A196199(n): return n-(t:=isqrt(n))*(t+1) # Chai Wah Wu, Aug 04 2022

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

A381233 Concatenate the sequences S(k) = [0, 1, -1, ..., k, -k] for k = 0, 1, ...

Original entry on oeis.org

0, 0, 1, -1, 0, 1, -1, 2, -2, 0, 1, -1, 2, -2, 3, -3, 0, 1, -1, 2, -2, 3, -3, 4, -4, 0, 1, -1, 2, -2, 3, -3, 4, -4, 5, -5, 0, 1, -1, 2, -2, 3, -3, 4, -4, 5, -5, 6, -6, 0, 1, -1, 2, -2, 3, -3, 4, -4, 5, -5, 6, -6, 7, -7, 0, 1, -1, 2, -2, 3, -3, 4, -4, 5, -5, 6, -6, 7, -7, 8, -8, 0, 1, -1, 2, -2, 3, -3, 4, -4, 5, -5, 6, -6, 7, -7, 8, -8, 9, -9

Offset: 0

N. J. A. Sloane, Mar 01 2025 [Suggested by Franklin T. Adams-Watters, Sep 21 2011]

Paolo Xausa, Table of n, a(n) for n = 0..10200 (k = 0..100)

Cf. A196199, A381232.

Flatten[Table[(-1)^j*Floor[j/2], {k, 0, 10}, {j, 2*k + 1}]] (* Paolo Xausa, Mar 01 2025 *)

cp's OEIS Frontend

A196199 Count up from -n to n for n = 0, 1, 2, ... .

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