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A196199 Count up from -n to n for n = 0, 1, 2, ... . 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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

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.

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

EXAMPLE

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

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

PROG

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

(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

CROSSREFS

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

Cf. A004737.

Sequence in context: A106509 A228110 A255175 * A053615 A002819 A037834

Adjacent sequences:  A196196 A196197 A196198 * A196200 A196201 A196202

KEYWORD

sign,tabf,easy,frac,look

AUTHOR

Franklin T. Adams-Watters, Sep 29 2011

STATUS

approved

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Last modified December 7 11:34 EST 2016. Contains 278874 sequences.