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A209278 Second inverse function (numbers of rows) for pairing function A185180. 4
1, 2, 1, 2, 3, 1, 3, 2, 4, 1, 3, 4, 2, 5, 1, 4, 3, 5, 2, 6, 1, 4, 5, 3, 6, 2, 7, 1, 5, 4, 6, 3, 7, 2, 8, 1, 5, 6, 4, 7, 3, 8, 2, 9, 1, 6, 5, 7, 4, 8, 3, 9, 2, 10, 1, 6, 7, 5, 8, 4, 9, 3, 10, 2, 11, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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)+3)/2) + floor(A002260(n)/2)*(-1)^(A002260(n)+A003056(n)).

a(n)= floor((t+3)/2)+ floor(i/2)*(-1)^(i+t),

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

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

EXAMPLE

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

1...2...2...3...3...4...4...5...

1...3...2...4...3...5...4...6...

1...4...2...5...3...6...4...7...

1...5...2...6...3...7...4...8...

1...6...2...7...3...8...4...9...

1...7...2...8...3...9...4..10...

1...8...2...9...3..10...4..11...

. . .

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

1;

2, 1;

2, 3, 1;

3, 2, 4, 1;

3, 4, 2, 5, 1;

4, 3, 5, 2, 6, 1;

4, 5, 3, 6, 2, 7, 1;

5, 4, 6, 3, 7, 2, 8, 1;

. . .

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, ... 2, r, 1.

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

MATHEMATICA

T[r_, s_] := If[OddQ[s], (s+1)/2, r + s/2];

Table[T[r-s+1, s], {r, 1, 11}, {s, r, 1, -1}] // Flatten (* Jean-Fran├žois Alcover, Nov 19 2019 *)

PROG

(Python)

t=int((math.sqrt(8*n-7) - 1)/ 2)

i=n-t*(t+1)/2

result=int((t+3)/2)+int(i/2)*(-1)**(i+t)

(PARI) T(r, s)=s\2+if(bittest(s, 0), 1, r) \\ - M. F. Hasler, Jan 15 2013

CROSSREFS

Cf. A185180, A092542, A092543.

Sequence in context: A231516 A048793 A249783 * A326921 A286343 A075106

Adjacent sequences:  A209275 A209276 A209277 * A209279 A209280 A209281

KEYWORD

nonn,tabl

AUTHOR

Boris Putievskiy, Jan 15 2013

STATUS

approved

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Last modified April 5 00:43 EDT 2020. Contains 333238 sequences. (Running on oeis4.)