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A056011 Unique triangle such that (1) every positive integer occurs exactly once; (2) row n consists of n consecutive numbers; (3) odd-numbered rows are increasing; and (4) even-numbered rows are decreasing. 12
1, 3, 2, 4, 5, 6, 10, 9, 8, 7, 11, 12, 13, 14, 15, 21, 20, 19, 18, 17, 16, 22, 23, 24, 25, 26, 27, 28, 36, 35, 34, 33, 32, 31, 30, 29, 37, 38, 39, 40, 41, 42, 43, 44, 45, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Self-inverse permutation of the natural numbers.

Mirror image of triangle in A056023. - Philippe Deléham, Apr 04 2009

a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers. - Boris Putievskiy, Dec 24 2012

For generalizations see A218890, A213927. - Boris Putievskiy, Mar 10 2013

LINKS

Reinhard Zumkeller, Rows n = 1..125 of table, flattened

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

Eric W. Weisstein, MathWorld: Pairing functions

Index entries for sequences that are permutations of the natural numbers

FORMULA

a(n) = ((i+j-1)*(i+j-2)+((-1)^t+1)*i - ((-1)^t-1)*j)/2, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 24 2012

EXAMPLE

The start of the sequence as table:

   1,  3,  4, 10, 11, 21, ...

   2,  5,  9, 12, 20, 23, ...

   6,  8, 13, 19, 24, 34, ...

   7, 14, 18, 25, 33, 40, ...

  15, 17, 26, 32, 41, 51, ...

  ...

Enumeration by boustrophedonic ("ox-plowing") diagonal method. - Boris Putievskiy, Dec 24 2012

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

   1;

   3,  2;

   4,  5,  6;

  10,  9,  8,  7;

  11, 12, 13, 14, 15;

  ...

MAPLE

A056011 := proc(n, k)

        if type(n, 'even') then

                A131179(n)-k+1 ;

        else

                A131179(n)+k-1 ;

        end if;

end proc: # R. J. Mathar, Sep 05 2012

MATHEMATICA

Flatten[If[EvenQ[Length[#]], Reverse[#], #]&/@Table[c=(n(n+1))/2; Range[ c-n+1, c], {n, 20}]] (* Harvey P. Dale, Mar 25 2012 *)

PROG

(Haskell)

a056011 n = a056011_tabl !! (n-1)

a056011_list = concat a056011_tabl

a056011_tabl = ox False a000027_tabl where

  ox turn (xs:xss) = (if turn then reverse xs else xs) : ox (not turn) xss

a056011_row n = a056011_tabl !! (n-1)

-- Reinhard Zumkeller, Nov 08 2013

CROSSREFS

Cf. A079826, A131179 (first column), A218890, A213927.

Cf. A000027, A038722.

Sequence in context: A113001 A036812 A039906 * A117123 A116966 A182187

Adjacent sequences:  A056008 A056009 A056010 * A056012 A056013 A056014

KEYWORD

nonn,tabl,nice,easy

AUTHOR

Clark Kimberling, Aug 01 2000

STATUS

approved

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Last modified February 19 16:28 EST 2018. Contains 299356 sequences. (Running on oeis4.)