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A209279 First inverse function (numbers of rows) for pairing function A185180. 5
1, 1, 2, 2, 1, 3, 2, 3, 1, 4, 3, 2, 4, 1, 5, 3, 4, 2, 5, 1, 6, 4, 3, 5, 2, 6, 1, 7, 4, 5, 3, 6, 2, 7, 1, 8, 5, 4, 6, 3, 7, 2, 8, 1, 9, 5, 6, 4, 7, 3, 8, 2, 9, 1, 10, 6, 5, 7, 4, 8, 3, 9, 2, 10, 1, 11, 6, 7, 5, 8, 4, 9, 3, 10, 2, 11, 1, 12, 7, 6, 8, 5, 9, 4, 10, 3, 11, 2, 12, 1, 13 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

Boris Putievskiy, Rows n = 1..140 of triangle, flattened

Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Eric Weisstein's World of Mathematics, Pairing functions

FORMULA

a(n) = floor((A003056(n)+2)/2)+ floor(A002260(n)/2)*(-1)^(A002260(n)+A003056(n)+1).

a(n) = |A128180(n)|.

a(n) = floor((t+2)/2) + floor(i/2)*(-1)^(i+t+1), where t=floor((-1+sqrt(8*n-7))/2), i=n-t*(t+1)/2.

T(r,2s)=s, T(r,2s-1)= r+s-1.(When read as table T(r,s) by antidiagonals.)

T(n,k) = ceiling((n + (-1)^(n-k)*k)/2) = (n+k)/2 if n-k even, otherwise (n-k+1)/2. - M. F. Hasler, May 30 2020

EXAMPLE

The start of the sequence as table T(r,s) r,s >0 read by antidiagonals:

  1...1...2...2...3...3...4...4...

  2...1...3...2...4...3...5...4...

  3...1...4...2...5...3...6...4...

  4...1...5...2...6...3...7...4...

  5...1...6...2...7...3...8...4...

  6...1...7...2...8...3...9...4...

  7...1...8...2...9...3..10...4...

  ...

The start of the sequence as triangle array read by rows:

  1;

  1, 2;

  2, 1, 3;

  2, 3, 1, 4;

  3, 2, 4, 1, 5;

  3, 4, 2, 5, 1, 6;

  4, 3, 5, 2, 6, 1, 7;

  4, 5, 3, 6, 2, 7, 1, 8;

  ...

Row number r contains permutation numbers form 1 to r.

If r is odd (r+1)/2, (r+1)/2-1, (r+1)/2+1,...r-1, 1, r.

If r is even r/2, r/2+1, r/2-1, ... r-1, 1, r.

MATHEMATICA

T[n_, k_] := Abs[(2*k - 1 + (-1)^(n - k)*(2*n + 1))/4];

Table[T[n, k], {n, 1, 15}, {k, 1, n}] // Flatten (* Jean-Fran├žois Alcover, Jun 14 2018, after Andrew Howroyd *)

PROG

(PARI) T(n, k)=abs((2*k-1+(-1)^(n-k)*(2*n+1))/4) \\ Andrew Howroyd, Dec 31 2017

(Python) # Edited by M. F. Hasler, May 30 2020

def a(n):

   t = int((math.sqrt(8*n-7) - 1)/2);

   i = n-t*(t+1)/2;

   return int(t/2)+1+int(i/2)*(-1)**(i+t+1)

CROSSREFS

Cf. A185180, A128180, A092542, A092543, A209278.

Sequence in context: A002947 A241605 A128180 * A074754 A322529 A329949

Adjacent sequences:  A209276 A209277 A209278 * A209280 A209281 A209282

KEYWORD

nonn,tabl

AUTHOR

Boris Putievskiy, Jan 15 2013

EXTENSIONS

Data corrected by Andrew Howroyd, Dec 31 2017

STATUS

approved

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Last modified July 15 02:31 EDT 2020. Contains 335762 sequences. (Running on oeis4.)