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A056023 Unique triangle such that (1) every positive integer occurs exactly once; (2) row n consists of n consecutive numbers; (3) odd-numbered rows are decreasing; (4) even-numbered rows are increasing; and (5) column 1 is increasing. 9
1, 2, 3, 6, 5, 4, 7, 8, 9, 10, 15, 14, 13, 12, 11, 16, 17, 18, 19, 20, 21, 28, 27, 26, 25, 24, 23, 22, 29, 30, 31, 32, 33, 34, 35, 36, 45, 44, 43, 42, 41, 40, 39, 38, 37, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 66, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Self-inverse permutation of the natural numbers.

T(2*n-1,1) = A000217(2*n-1) = T(2*n,1) - 1; T(2*n,4*n) = A000217(2*n) = T(2*n+1,4*n+1) - 1. - Reinhard Zumkeller, Apr 25 2004

Mirror image of triangle in A056011. - Philippe Deléham, Apr 04 2009

From Clark Kimberling, Feb 03 2011: (Start)

When formatted as a rectangle R, for m > 1, the numbers n-1 and n+1 are neighbors (row, column, or diagonal) of R.

R(n,k) = n + (k+n-2)(k+n-1)/2 if n+k is odd;

R(n,k) = k + (n+k-2)(n+k-1)/2 if n+k is even.

Northwest corner:

   1,  2,  6,  7, 15, 16, 28

   3,  5,  8, 14, 17, 27, 30

   4,  9, 13, 18, 26, 31, 43

  10, 12, 19, 25, 32, 42, 49

  11, 20, 24, 33, 41, 50, 62

  (End)

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

Ivan Neretin, Table of n, a(n) for n = 1..5050

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

Eric Weisstein's MathWorld, Pairing functions

Index entries for sequences that are permutations of the natural numbers

FORMULA

T(n, k) = (n^2 - (n - 2*k)*(-1)^(n mod 2))/2 + n mod 2. - Reinhard Zumkeller, Apr 25 2004

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

EXAMPLE

From Philippe Deléham, Apr 04 2009 (Start)

Triangle begins:

  1;

  2,   3;

  6,   5,  4;

  7,   8,  9, 10;

  15, 14, 13, 12, 11;

  ...

(End)

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

MATHEMATICA

(* As a rectangle: *)

r[n_, k_] := n + (k + n - 2) (k + n - 1)/2/; OddQ[n + k];

r[n_, k_] := k + (n + k - 2) (n + k - 1)/2/; EvenQ[n+k];

TableForm[Table[f[n, k], {n, 1, 10}, {k, 1, 15}]]

Table[r[n - k + 1, k], {n, 14}, {k, n, 1, -1}]//Flatten

(* Clark Kimberling, Feb 03 2011 *)

CROSSREFS

Cf. A056011, A218890, A213927.

Sequence in context: A130686 A213927 A222241 * A133259 A120067 A089843

Adjacent sequences:  A056020 A056021 A056022 * A056024 A056025 A056026

KEYWORD

nonn,tabl,easy

AUTHOR

Clark Kimberling, Aug 01 2000

STATUS

approved

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Last modified September 25 00:48 EDT 2020. Contains 337333 sequences. (Running on oeis4.)