A218890 -id:A218890 - cp's OEIS frontend

A056023 The positive integers written as a triangle, where row n is written from right to left if n is odd and otherwise from left to right.

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

Offset: 1

Clark Kimberling, Aug 01 2000

A 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.
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

			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

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

Cf. A056011, A218890, A213927.

(* 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 *)
Module[{nn=15},If[OddQ[Length[#]],Reverse[#],#]&/@TakeList[Range[ (nn(nn+1))/2],Range[nn]]]//Flatten (* Harvey P. Dale, Feb 08 2022 *)

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

Name edited by Andrey Zabolotskiy, Apr 16 2023

A056011 Enumeration of natural numbers by the boustrophedonic diagonal method.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Aug 01 2000

A 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.
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

			The start of the sequence as a 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;
  ...

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

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

Cf. A000027, A038722.

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

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

Flatten[If[EvenQ[Length[#]],Reverse[#],#]&/@Table[c=(n(n+1))/2;Range[ c-n+1,c],{n,20}]] (* Harvey P. Dale, Mar 25 2012 *)
With[{nn=20},{#[[1]],Reverse[#[[2]]]}&/@Partition[ TakeList[ Range[ (nn(nn+1))/2],Range[nn]],2]//Flatten] (* Harvey P. Dale, Oct 05 2021 *)

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

New name from Peter Luschny, Apr 15 2023, based on Boris Putievskiy's comment

A213927 T(n,k) = (z(z-1)-(-1+(-1)^(z^2 mod 3))n+(1+(-1)^(z^2 mod 3))*k)/2, where z=n+k-1; n, k > 0, read by antidiagonals.

Original entry on oeis.org

1, 2, 3, 6, 5, 4, 7, 8, 9, 10, 11, 12, 13, 14, 15, 21, 20, 19, 18, 17, 16, 22, 23, 24, 25, 26, 27, 28, 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, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 78

Offset: 1

Boris Putievskiy, Mar 06 2013

Self-inverse permutation of the natural numbers.
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.
In general, let b(z) be a sequence of integers and denote number of antidiagonal table T(n,k) by z=n+k-1. Natural numbers placed in table T(n,k) by antidiagonals. The order of placement - by  antidiagonal downwards, if b(z) is odd; by  antidiagonal upwards, if b(z) is even. T(n,k) read by antidiagonals downwards. For A218890 -- the order of placement -- at the beginning m antidiagonals downwards, next m antidiagonals upwards and so on - b(z)=floor((z+m-1)/m). For this sequence b(z)=z^2 mod 3. (This comment should be edited for clarity, Joerg Arndt, Dec 11 2014)

			The start of the sequence as table.
The direction of the placement denoted by ">" and  "v".
.v.....v       v...v        v....v
.1.....2...6...7..11...21...22...29...45...
.3.....5...8..12..20...23...30...44...47...
>4.....9..13..19..24...31...43...48...58...
.10...14..18..25..32...42...49...59...75...
.15...17..26..33..41...50...60...74...83...
>16...27..34..40..51...61...73...84...97...
.28...35..39..52..62...72...85...98..114...
.36...38..53..63..71...86...99..113..128...
>37...54..64..70..87..100..112..129..145...
...
The start of the sequence as triangle array read by rows:
   1;
   2,  3;
   6,  5,  4;
   7,  8,  9, 10;
  11, 12, 13, 14, 15;
  21, 20, 19, 18, 17, 16;
  22, 23, 24, 25, 26, 27, 28;
  29, 30, 31, 32, 33, 34, 35, 36;
  45, 44, 43, 42, 41, 40, 39, 38, 37;
  ...
Row r consists of r consecutive numbers from r*r/2-r/2+1 to r*r/2+r.
If r is not divisible by 3, rows are increasing.
If r is     divisible by 3, rows are decreasing.

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

Cf. A218890, A056011, A056023, A130196, A011655, A001651, A008585.

T[n_, k_] := With[{z = n + k - 1}, (z*(z - 1) - (-1 + (-1)^Mod[z^2, 3])*n + (1 + (-1)^Mod[z^2, 3])*k)/2];
Table[T[n - k + 1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jul 22 2018 *)

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
z=i+j-1
result=(z*(z-1)-(-1+(-1)**(z**2%3))*i+(1+(-1)**(z**2%3))*j)/2

For the general case.
T(n,k) = (z*(z-1)-(-1+(-1)^b(z))*n+(1+(-1)^b(z))*k)/2, where z=n+k-1 (as a table).
a(n) = (z*(z-1)-(-1+(-1)^b(z))*i+(1+(-1)^b(z))*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), z=i+j-1 (as a linear sequence).
For this sequence b(z)=z^2 mod 3.
T(n,k) = (z*(z-1)-(-1+(-1)^(z^2 mod 3))*n+(1+(-1)^(z^2 mod 3))*k)/2, where z=n+k-1 (as a table).
a(n) = (z*(z-1)-(-1+(-1)^(z^2 mod 3))*i+(1+(-1)^(z^2 mod 3))*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), z=i+j-1 (as linear sequence).

cp's OEIS Frontend

A056023 The positive integers written as a triangle, where row n is written from right to left if n is odd and otherwise from left to right.

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A056011 Enumeration of natural numbers by the boustrophedonic diagonal method.

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A213927 T(n,k) = (z(z-1)-(-1+(-1)^(z^2 mod 3))n+(1+(-1)^(z^2 mod 3))*k)/2, where z=n+k-1; n, k > 0, read by antidiagonals.

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cp's OEIS Frontend

A056023 The positive integers written as a triangle, where row n is written from right to left if n is odd and otherwise from left to right.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A056011 Enumeration of natural numbers by the boustrophedonic diagonal method.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Formula

Extensions

A213927 T(n,k) = (z*(z-1)-(-1+(-1)^(z^2 mod 3))*n+(1+(-1)^(z^2 mod 3))*k)/2, where z=n+k-1; n, k > 0, read by antidiagonals.

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Mathematica

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Formula

A213927 T(n,k) = (z(z-1)-(-1+(-1)^(z^2 mod 3))n+(1+(-1)^(z^2 mod 3))*k)/2, where z=n+k-1; n, k > 0, read by antidiagonals.