A003057 -id:A003057 - cp's OEIS frontend

A349083 The number of three-term Egyptian fractions of rational numbers x/y, 0 < x/y < 1, ordered as below. The sequence is the number of (p,q,r) such that x/y = 1/p + 1/q + 1/r where p, q, and r are integers with p < q < r.

6, 15, 5, 22, 6, 3, 30, 9, 7, 2, 45, 15, 6, 5, 1, 36, 14, 6, 5, 3, 1, 62, 22, 16, 6, 5, 3, 2, 69, 21, 15, 4, 9, 5, 2, 1, 84, 30, 15, 9, 6, 7, 2, 2, 1, 56, 22, 13, 7, 3, 5, 2, 0, 0, 0, 142, 45, 22, 15, 12, 6, 9, 5, 3, 1, 2, 53, 17, 8, 4, 5, 1, 6, 3, 1, 1, 1, 0, 124, 36, 27, 14, 18, 6, 6, 5, 2, 3, 1, 1, 0

Offset: 1

Jud McCranie, Nov 09 2021

The sequence are the terms in a triangle, where the rows correspond to the denominator of the rational number (starting with row 2, column 1) and the columns correspond to the numerators:
            x = 1   2   3   4   5   Rationals x/y:
Row 1: (y=2)    6                   1/2
Row 2: (y=3)   15,  5               1/3, 2/3
Row 3: (y=4)   22,  6,  3           1/4, 2/4, 3/4
Row 4: (y=5)   30,  9,  7,  2       1/5, 2/5, 3/5, 4/5
Row 5: (y=6)   45, 15,  6,  5,  1   1/6, 2/6, 3/6, 4/6, 5/6
Alternatively, order the rational numbers, x/y, 0 < x/y < 1, in this order: 1/2, 1/3, 2/3, 1/4, 2/4, 3/4, 1/5, 2/5, ... The numerators of the n-th rational number are A002260(n) and the denominators are A003057(n).

			The sixth rational number is 3/4;
  3/4 = 1/2 + 1/5 + 1/20
      = 1/2 + 1/6 + 1/12
      = 1/3 + 1/4 + 1/5,
so a(6)=3.

Jud McCranie, Table of n, a(n) for n = 1..990

Cf. A002260, A003057, A349082.

Columns x=1..5: A227610, A227611, A075785, A073101, A075248.

Efrac3(x,y)=sum(p=if(y%x,y\x,y\x+1),3*y\x, my(N=x/y-1/p); sum(q=max(if(numerator(N)==1,1\N+1,1\N),p+1),2\N, my(M=N-1/q,r=1/M); type(r)=="t_INT" && qCharles R Greathouse IV, Nov 09 2021

A366727 2-tone chromatic number of a maximal outerplanar graph with maximum degree n.

Original entry on oeis.org

4, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15

Offset: 1

Allan Bickle, Oct 17 2023

nonn

The 2-tone chromatic number of a graph G is the smallest number of colors for which G has a coloring where every vertex has two distinct colors, no adjacent vertices have a common color, and no pair of vertices at distance 2 have two common colors.

a(n) is also the 2-tone chromatic number of a fan with n+1 vertices.

			The fan with 11 vertices has a path colored 12-34-15-23-45-13-24-35-14-25 joined to a vertex colored 67, so a(10) = 7.

Allan Bickle, 2-Tone coloring of joins and products of graphs, Congr. Numer. 217 (2013), 171-190.
Allan Bickle, 2-Tone Coloring of Chordal and Outerplanar Graphs, Australas. J. Combin. 87 1 (2023) 182-197.
Allan Bickle and B. Phillips, t-Tone Colorings of Graphs, Utilitas Math, 106 (2018) 85-102.
D. W. Cranston and H. LaFayette, The t-tone chromatic number of classes of sparse graphs, Australas. J. Combin. 86 (2023), 458-476.
N. Fonger, J. Goss, B. Phillips, and C. Segroves, Math 6450: Final Report, Group #2 Study Project, 2009.

Cf. A350361 (trees), A350362 (cycles), A350715 (wheels), A366728 (cycle squared).

Cf. A003057, A351120 (pair coloring).

a(n) = ceiling(sqrt(2*n + 1/4) + 5/2) for n > 6.

A366728 2-tone chromatic number of the square of a cycle with n vertices.

Original entry on oeis.org

6, 8, 10, 9, 7, 8, 8, 8, 8, 7, 8, 7, 7, 7, 8, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 3

Allan Bickle, Oct 17 2023

nonn

The 2-tone chromatic number of a graph G is the smallest number of colors for which G has a coloring where every vertex has two distinct colors, no adjacent vertices have a common color, and no pair of vertices at distance 2 have two common colors.

The square of a cycle is formed by adding edges between all vertices at distance 2 in the cycle.

			The colorings for (broken) cycles with orders 7 through 13 are shown below.
  -12-34-56-71-23-45-67-
  -12-34-56-78-13-24-57-68-
  -12-34-56-17-23-45-16-37-58-
  -12-34-56-71-23-68-15-24-38-57-
  -12-34-56-17-24-36-58-14-26-38-57-
  -12-34-56-71-32-54-16-37-52-14-36-57-
  -12-34-56-71-32-54-16-37-58-14-32-57-68-

Allan Bickle, 2-Tone coloring of joins and products of graphs, Congr. Numer. 217 (2013), 171-190.
Allan Bickle, 2-Tone Coloring of Chordal and Outerplanar Graphs, Australas. J. Combin. 87 1 (2023) 182-197.
Allan Bickle and B. Phillips, t-Tone Colorings of Graphs, Utilitas Math, 106 (2018) 85-102.
D. W. Cranston and H. LaFayette, The t-tone chromatic number of classes of sparse graphs, Australas. J. Combin. 86 (2023), 458-476.
N. Fonger, J. Goss, B. Phillips, and C. Segroves, Math 6450: Final Report, Group #2 Study Project, 2009.

Cf. A350361 (trees), A350362 (cycles), A350715 (wheels), A366727 (MOPs).

Cf. A003057, A351120 (pair coloring).

a(n) = 7 for all n>17.

A108872 Sums of ordinal references for a triangular table read by columns, top to bottom.

Original entry on oeis.org

2, 3, 4, 4, 5, 6, 5, 6, 7, 8, 6, 7, 8, 9, 10, 7, 8, 9, 10, 11, 12, 8, 9, 10, 11, 12, 13, 14, 9, 10, 11, 12, 13, 14, 15, 16, 10, 11, 12, 13, 14, 15, 16, 17, 18, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22

Offset: 1

Andrew S. Plewe, Jul 13 2005

The ordinal references (i,j) for a triangular table are arranged as follows:
  (1,1) (2,1) (3,1)
  ..... (2,2) (3,2)
  ........... (3,3)
The sequence comprises the sum of each reference in each column, read top to bottom. A similar sequence is A003057, which consists of the sums of the ordinal references for an array read by antidiagonals.
Subtriangle of triangle in A051162. - Philippe Deléham, Mar 26 2013
First 9 rows coincide with triangle A248110; T(n,k) = A002260(n,k) + n; T(2*n-1,n) = A016789(n-1). - Reinhard Zumkeller, Oct 01 2014

			a(1) = (1,1) = 1 + 1 = 2
a(2) = (2,1) = 2 + 1 = 3
a(3) = (2,2) = 2 + 2 = 4
a(4) = (3,1) = 3 + 1 = 4, etc.
Triangle begins:
  2
  3, 4
  4, 5, 6
  5, 6, 7, 8
  6, 7, 8, 9, 10
  7, 8, 9, 10, 11, 12
  8, 9, 10, 11, 12, 13, 14
  9, 10, 11, 12, 13, 14, 15, 16
  ... - _Philippe Deléham_, Mar 26 2013

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
NRICH discussion thread, Floor Function Identity. [Via Wayback Machine]

Cf. A003057.
Cf. A002024, A002260.
Cf. A005408, A005843.
Cf. A016789 (central terms), A248110.

a108872 n k = a108872_tabl !! (n-1) !! (k-1)
a108872_row n = a108872_tabl !! (n-1)
a108872_tabl = map (\x -> [x + 1 .. 2 * x]) [1..]
-- Reinhard Zumkeller, Oct 01 2014

Flatten[ Table[i + j, {j, 1, 12}, {i, 1, j}]] (* Jean-François Alcover, Oct 07 2011 *)

from math import isqrt
def A108872(n): return n+((r:=(m:=isqrt(k:=n<<1))+(k>m*(m+1)))*(3-r)>>1) # Chai Wah Wu, Nov 08 2024

a(n) = a(i, j) = i + j
a(n) = A002024(n) + A002260(n) = floor(1/2 + sqrt(2n)) + n - (m(m+1)/2) + 1, where m = floor((sqrt(8n+1) - 1) / 2 ). The floor function may be computed directly by using the expression floor(x) = x + (arctan(cot(Pi*x)) / Pi) - 1/2 (equation from nrich.maths.org, see links).
Sum_{k=0..n} T(n,k) = A005449(n+1). - Philippe Deléham, Mar 26 2013

Offset changed by Reinhard Zumkeller, Oct 01 2014

A140978 Repeat (n+1)^2 n times.

Original entry on oeis.org

4, 9, 9, 16, 16, 16, 25, 25, 25, 25, 36, 36, 36, 36, 36, 49, 49, 49, 49, 49, 49, 64, 64, 64, 64, 64, 64, 64, 81, 81, 81, 81, 81, 81, 81, 81, 100, 100, 100, 100, 100, 100, 100, 100, 100, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121

Offset: 1

Paul Curtz, Aug 17 2008

See A093995.
Frenicle writes the entries in the form a(n) = A055096(n)-A133819(n), with the flattened index view of A133819: 4=5-1, 9=10-1, 9=13-4, 16=17-1, 16=20-4, 16=25-9 etc.
Also triangle T(n, k) = (n+1)^2, 1<=k<=n. - Michel Marcus, Feb 03 2013

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
M. de Frenicle, Methode pour trouver la solutions des problemes par les exclusions, in: Divers ouvrages des mathematiques et de physique par messieurs de l'academie royale des sciences, (1693) pp 1-44, table page 11.

Cf. A000290.

a140978 n k = a140978_tabl !! (n-1) !! (k-1)
a140978_row n = a140978_tabl !! (n-1)
a140978_tabl = map snd $ iterate
               (\(i, xs@(x:_)) -> (i + 2, map (+ i) (x:xs))) (5, [4])
-- Reinhard Zumkeller, Mar 23 2013

Table[PadRight[{},n,(n+1)^2],{n,10}]//Flatten (* Harvey P. Dale, Oct 10 2019 *)

from math import isqrt
def A140978(n): return ((m:=isqrt(k:=n<<1))+(k>m*(m+1))+1)**2 # Chai Wah Wu, Nov 07 2024

a(n)=(A003057(n+1))^2. - R. J. Mathar, Aug 25 2008

A349084 The number of four-term Egyptian fractions of rational numbers, x/y, 0 < x/y < 1, ordered as below. The sequence is the number of (p,q,r,s) such that x/y = 1/p + 1/q + 1/r + 1/s where p, q, r, and s are integers with p < q < r < s.

Original entry on oeis.org

71, 272, 61, 586, 71, 27, 978, 275, 122, 18, 1591, 272, 71, 61, 17, 1865, 564, 130, 145, 31, 18, 3115, 586, 478, 71, 85, 27, 17, 3772, 1079, 272, 109, 218, 61, 23, 11, 4964, 978, 461, 275, 71, 122, 39, 18, 9, 4225, 1208, 641, 400, 59, 174, 37, 16, 5, 3, 8433, 1591, 586, 272, 214, 71, 172, 61, 27, 17, 12

Offset: 1

Jud McCranie, Nov 11 2021

The sequence are the terms in a triangle, where the rows correspond to the denominator of the rational number (starting with row 2, column 1) and the columns correspond to the numerators:
                    x= 1    2    3    4    5   Rationals x/y:
Row 1: (y=2)    71                       1/2
Row 2: (y=3)   272,  61                  1/3, 2/3
Row 3: (y=4)   586,  71,  27             1/4, 2/4, 3/4
Row 4: (y=5)   978, 275, 122,  18        1/5, 2/5, 3/5, 4/5
Row 5: (y=6)  1591, 272,  71,  61,  17   1/6, 2/6, 3/6, 4/6, 5/6
Alternatively, order the rational numbers, x/y, 0 < x/y < 1, in this order: 1/2, 1/3, 2/3, 1/4, 2/4, 3/4, 1/5, 2/5, ... The numerators of the n-th rational number are A002260(n) and the denominators are A003057(n).
Column 1 is A241883.

			The 10th rational number under this ordering is 4/5; 4/5 has 18 representations as the sum of four distinct unit fractions, so a(10) = 18:
4/5 = 1/2 + 1/4 + 1/21 + 1/420
   = 1/2 + 1/4 + 1/22 + 1/220
   ... 15 solutions omitted
   = 1/3 + 1/5 + 1/6 + 1/10

Jud McCranie, Table of n, a(n) for n = 1..990

Cf. A002260, A003057, A349082, A349083, A241883.

A106448 Table of (x+y)/gcd(x,y) where (x,y) runs through the pairs (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), ...

Original entry on oeis.org

2, 3, 3, 4, 2, 4, 5, 5, 5, 5, 6, 3, 2, 3, 6, 7, 7, 7, 7, 7, 7, 8, 4, 8, 2, 8, 4, 8, 9, 9, 3, 9, 9, 3, 9, 9, 10, 5, 10, 5, 2, 5, 10, 5, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 6, 4, 3, 12, 2, 12, 3, 4, 6, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 7, 14, 7, 14, 7, 2, 7, 14, 7, 14, 7, 14

Offset: 1

Antti Karttunen, May 21 2005

Can also be viewed as a triangular table T(n,k) (n>=1, 1<=k<=n) read by rows: T(1,1); T(2,1), T(2,2); T(3,1), T(3,2), T(3,3); T(4,1), T(4,2), T(4,3), T(4,4); ... where T(n,k) gives the least value v>0 such that v*k = 0 modulo n+1, i.e., in other words, T(n,k) = (n+1)/gcd(n+1,k).

			The top left corner of the square array is:
   2  3  4  5  6  7  8  9 10 11 ...
   3  2  5  3  7  4  9  5 11 ...
   4  5  2  7  8  3 10 11 ...
   5  3  7  2  9  5 11 ...
   6  7  8  9  2 11 ...
   7  4  3  5 11 ...
   8  9 10 11 ...
   9  5 11 ...
  10 11 ...
  11 ...

GF(2)[X] analog: A106449. Row 1 is n+1, row 2 is LEFT(LEFT(LEFT(A026741))), row 3 is LEFT^4(A051176). Essentially the same as A054531, but without its right-hand edge of all-1's.

Equals A003057/A003989.

T(n, k) = numerator((n+k)/n) = numerator((n+k)/k). - Michel Marcus, Dec 29 2013

A127739 Triangle read by rows, in which row n contains the triangular number T(n) = A000217(n) repeated n times; smallest triangular number greater than or equal to n.

Original entry on oeis.org

1, 3, 3, 6, 6, 6, 10, 10, 10, 10, 15, 15, 15, 15, 15, 21, 21, 21, 21, 21, 21, 28, 28, 28, 28, 28, 28, 28, 36, 36, 36, 36, 36, 36, 36, 36, 45, 45, 45, 45, 45, 45, 45, 45, 45, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66

Offset: 1

Gary W. Adamson, Jan 27 2007

Seen as a sequence, these are the triangular numbers applied to the Kruskal-Macaulay function A123578. - Peter Luschny, Oct 29 2022

			First few rows of the triangle are:
   1;
   3,  3;
   6,  6,  6;
  10, 10, 10, 10;
  15, 15, 15, 15, 15;
  ...

Reinhard Zumkeller, Rows n=1..100 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Cf. A000217, A000384, A002024, A002411 (row sums), A003057, A057944, A123578.

a127739 n k = a127739_tabl !! (n-1) !! (k-1)
a127739_row n = a127739_tabl !! (n-1)
a127739_tabl = zipWith ($) (map replicate [1..]) $ tail a000217_list
-- Reinhard Zumkeller, Feb 03 2012, Mar 18 2011

A127739 := proc(n) local t, s; t := 1; s := 0;
while t <= n do s := s + 1; t := t + s od; s*(1 + s)/2 end:
seq(A127739(n), n = 1..66); # Peter Luschny, Oct 29 2022

Table[n(n+1)/2,{n,100},{n}]//Flatten (* Zak Seidov, Mar 19 2011 *)

A127739=n->binomial((sqrtint(8*n)+3)\2,2) \\ M. F. Hasler, Mar 09 2014

from math import isqrt
def A127739(n): return (r:=(m:=isqrt(k:=n<<1))+(k>m*(m+1)))*(r+1)>>1 # Chai Wah Wu, Nov 07 2024

Central terms: T(2*n-1,n) = A000384(n). - Reinhard Zumkeller, Mar 18 2011
a(n) = A003057(n)*A002024(n)/2; a(n) = (t+2)*(t+1)/2, where t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Feb 08 2013
Sum_{n>=1} 1/a(n)^2 = 8 - 2*Pi^2/3. - Amiram Eldar, Aug 15 2022
a(n) = k(n)*(1 + k(n))/2 = A000217(A123578(n)), where k = A123578. - Peter Luschny, Oct 29 2022

Name edited by Michel Marcus, Apr 30 2020

A211394 T(n,k) = (k+n)(k+n-1)/2-(k+n-1)(-1)^(k+n)-k+2; n , k > 0, read by antidiagonals.

Original entry on oeis.org

1, 5, 6, 2, 3, 4, 12, 13, 14, 15, 7, 8, 9, 10, 11, 23, 24, 25, 26, 27, 28, 16, 17, 18, 19, 20, 21, 22, 38, 39, 40, 41, 42, 43, 44, 45, 29, 30, 31, 32, 33, 34, 35, 36, 37, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 80

Offset: 1

Boris Putievskiy, Feb 08 2013

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.
Enumeration table T(n,k). The order of the list:
T(1,1)=1;
T(1,3), T(2,2), T(3,1);
T(1,2), T(2,1);
. . .
T(1,n), T(2,n-1), T(3,n-2), ... T(n,1);
T(1,n-1), T(2,n-3), T(3,n-4),...T(n-1,1);
. . .
First row  matches with the elements antidiagonal {T(1,n), ... T(n,1)},
second row matches with the elements antidiagonal {T(1,n-1), ... T(n-1,1)}.
Table contains:
row  1 is alternation of elements A130883 and A096376,
row  2  accommodates elements A033816 in even places,
row  3  accommodates elements A100037 in odd  places,
row  5  accommodates elements A100038 in odd  places;
column 1 is alternation of elements A084849 and  A000384,
column 2 is alternation of elements A014106 and  A014105,
column 3 is alternation of elements A014107 and  A091823,
column 4 is alternation of elements A071355 and |A168244|,
column 5 accommodates elements A033537 in even places,
column 7 is alternation of elements A100040 and  A130861,
column 9 accommodates elements A100041 in even places;
the main diagonal is A058331,
diagonal  1, located above the main diagonal is  A001844,
diagonal  2, located above the main diagonal is  A001105,
diagonal  3, located above the main diagonal is  A046092,
diagonal  4, located above the main diagonal is  A056220,
diagonal  5, located above the main diagonal is  A142463,
diagonal  6, located above the main diagonal is  A054000,
diagonal  7, located above the main diagonal is  A090288,
diagonal  9, located above the main diagonal is  A059993,
diagonal 10, located above the main diagonal is |A147973|,
diagonal 11, located above the main diagonal is  A139570;
diagonal  1, located under the main diagonal is  A051890,
diagonal  2, located under the main diagonal is  A005893,
diagonal  3, located under the main diagonal is  A097080,
diagonal  4, located under the main diagonal is  A093328,
diagonal  5, located under the main diagonal is  A137882.

			The start of the sequence as table:
  1....5...2..12...7..23..16...
  6....3..13...8..24..17..39...
  4...14...9..25..18..40..31...
  15..10..26..19..41..32..60...
  11..27..20..42..33..61..50...
  28..21..43..34..62..51..85...
  22..44..35..63..52..86..73...
  . . .
The start of the sequence as triangle array read by rows:
  1;
  5,6;
  2,3,4;
  12,13,14,15;
  7,8,9,10,11;
  23,24,25,26,27,28;
  16,17,18,19,20,21,22;
  . . .
Row number r matches with r numbers segment {(r+1)*r/2-r*(-1)^(r+1)-r+2,... (r+1)*r/2-r*(-1)^(r+1)+1}.

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 Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A130883, A096376, A033816, A100037, A100038, A084849, A000384, A014106, A014105, A014107, A091823, A071355, A168244, A033537, A100040, A130861, A100041, A058331, A001844, A001105, A046092, A056220, A142463, A054000, A090288, A059993, A147973, A139570, A051890, A005893, A097080, A093328, A137882.

T[n_, k_] := (n+k)(n+k-1)/2 - (-1)^(n+k)(n+k-1) - k + 2;
Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Dec 06 2018 *)

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

T(n,k) = (k+n)*(k+n-1)/2-(k+n-1)*(-1)^(k+n)-k+2.
As linear sequence
a(n) = A003057(n)*A002024(n)/2- A002024(n)*(-1)^A003056(n)-A004736(n)+2.
a(n) = (t+2)*(t+1)/2 - (t+1)*(-1)^t-j+2, where j=(t*t+3*t+4)/2-n and t=int((math.sqrt(8*n-7) - 1)/ 2).

A213171 T(n,k) = ((k+n)^2 - 4k + 3 - (-1)^n - (k+n)(-1)^(k+n))/2; n, k > 0, read by antidiagonals.

Original entry on oeis.org

1, 4, 5, 2, 3, 6, 9, 10, 13, 14, 7, 8, 11, 12, 15, 18, 19, 22, 23, 26, 27, 16, 17, 20, 21, 24, 25, 28, 31, 32, 35, 36, 39, 40, 43, 44, 29, 30, 33, 34, 37, 38, 41, 42, 45, 48, 49, 52, 53, 56, 57, 60, 61, 64, 65, 46, 47, 50, 51, 54, 55, 58, 59, 62, 63, 66, 69

Offset: 1

Boris Putievskiy, Feb 14 2013

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.
Enumeration table T(n,k). The order of the list:
T(1,1) = 1;
T(1,3), T(2,2), T(1,2), T(2,1), T(3,1);
. . .
T(1,n), T(2,n-1), T(1,n-1), T(2,n-2), T(3,n-2), T(4,n-3)...T(n,1);
...
Descent by snake along two adjacent antidiagonals - step to the southwest, step to the north, step to the southwest, step to the south and so on. The length of each step is 1. Phase four steps is rotated 90 degrees counterclockwise and the mirror of the phase A211377.
Table contains the following:
row 1 is alternation of elements A130883 and A100037,
row 2 accommodates elements A033816 in even places;
column 1 is alternation of elements A000384 and A014106,
column 2 is alternation of elements A091823 and A071355,
column 4 accommodates elements A130861 in odd places;
main diagonal is alternation of elements A188135 and A033567,
diagonal 1, located above the main diagonal, accommodates elements A033585 in even places,
diagonal 2, located above the main diagonal, accommodates elements A139271 in odd places,
diagonal 3, located above the main diagonal, is alternation of elements A033566 and A194431.

			The start of the sequence as a table:
   1   4   2   9   7   8  16 ...
   5   3  10   8  19  17  32 ...
   6  13  11  22  20  35  33 ...
  14  12  23  21  36  34  53 ...
  15  26  24  39  37  56  54 ...
  27  25  40  38  57  55  78 ...
  28  43  41  60  58  81  79 ...
  ...
The start of the sequence as a triangle array read by rows:
   1
   4  5
   2  3  6
   9 10 13 14
   7  8 11 12 15
  18 19 22 23 26 27
  16 17 20 21 24 25 28
  ...
The start of the sequence as array read by rows, the length of row r is 4*r-3.
First 2*r-2 numbers are from the row number 2*r-2 of triangle array, located above.
Last 2*r-1 numbers are from the row number 2*r-1 of triangle array, located above.
   1
   4  5  2  3  6
   9 10 13 14  7  8 11 12 15
  18 19 22 23 26 27 16 17 20 21 24 25 28
  ...
Row number r contains permutation 4*r-3 numbers from 2*r*r-5*r+4 to 2*r*r-r:
2*r*r-5*r+6, 2*r*r-5*r+7, ..., 2*r*r-r-4, 2*r*r-r-3, 2*r*r-r.

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 Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A000384, A002260, A003056, A003057, A004736, A014106, A033566, A033567, A033585, A033816, A071355, A091823, A100037, A130861, A130883, A139271, A188135, A194431, A211377.

T:=(n,k)->((k+n)^2-4*k+3-(-1)^n-(k+n)*(-1)^(k+n))/2: seq(seq(T(k,n-k),k=1..n-1),n=1..13); # Muniru A Asiru, Dec 06 2018

T[n_, k_] := ((n+k)^2 - 4k + 3 - (-1)^n - (-1)^(n+k)(n+k))/2;
Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Dec 06 2018 *)

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

As a table:
T(n,k) = ((k+n)^2-4*k+3-(-1)^n-(k+n)*(-1)^(k+n))/2.
As a linear sequence:
a(n) = (A003057(n)^2-4*A004736(n)+3-(-1)^A002260(n)-A003057(n)*(-1)^A003056(n))/2;
a(n) = ((t+2)^2-4*j+3-(-1)^i-(t+2)*(-1)^t)/2, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).

cp's OEIS Frontend

A349083 The number of three-term Egyptian fractions of rational numbers x/y, 0 < x/y < 1, ordered as below. The sequence is the number of (p,q,r) such that x/y = 1/p + 1/q + 1/r where p, q, and r are integers with p < q < r.

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A366727 2-tone chromatic number of a maximal outerplanar graph with maximum degree n.

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A366728 2-tone chromatic number of the square of a cycle with n vertices.

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A108872 Sums of ordinal references for a triangular table read by columns, top to bottom.

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A140978 Repeat (n+1)^2 n times.

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A349084 The number of four-term Egyptian fractions of rational numbers, x/y, 0 < x/y < 1, ordered as below. The sequence is the number of (p,q,r,s) such that x/y = 1/p + 1/q + 1/r + 1/s where p, q, r, and s are integers with p < q < r < s.

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A106448 Table of (x+y)/gcd(x,y) where (x,y) runs through the pairs (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), ...

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A127739 Triangle read by rows, in which row n contains the triangular number T(n) = A000217(n) repeated n times; smallest triangular number greater than or equal to n.

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A211394 T(n,k) = (k+n)(k+n-1)/2-(k+n-1)(-1)^(k+n)-k+2; n , k > 0, read by antidiagonals.

A213171 T(n,k) = ((k+n)^2 - 4k + 3 - (-1)^n - (k+n)(-1)^(k+n))/2; n, k > 0, read by antidiagonals.