A033638 -id:A033638 - cp's OEIS frontend

A266464 Number of n X 2 binary arrays with rows and columns lexicographically nondecreasing and column sums nonincreasing.

1, 2, 4, 7, 12, 19, 29, 42, 59, 80, 106, 137, 174, 217, 267, 324, 389, 462, 544, 635, 736, 847, 969, 1102, 1247, 1404, 1574, 1757, 1954, 2165, 2391, 2632, 2889, 3162, 3452, 3759, 4084, 4427, 4789, 5170, 5571, 5992, 6434, 6897, 7382, 7889, 8419, 8972, 9549, 10150

Offset: 0

R. H. Hardin, Dec 29 2015

			Some solutions for n=4:
..0..0....0..0....0..1....0..0....0..1....0..0....1..1....0..1....0..0....0..1
..0..0....0..0....0..1....1..1....1..0....0..0....1..1....1..0....0..0....1..0
..0..1....0..0....1..0....1..1....1..1....1..1....1..1....1..0....0..0....1..0
..1..0....1..1....1..0....1..1....1..1....1..1....1..1....1..1....0..0....1..0

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n=1..210 from R. H. Hardin)

Column 2 of A266470.

Partial sums of A033638.

a:= proc(n) option remember;
     `if`(n<0, 0, 1+a(n-1)+floor(n^2/4))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 27 2023

a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) -a(n-5).
From Colin Barker, Mar 21 2018: (Start)
G.f.: (x^3-x+1)/((x+1)*(x-1)^4).
a(n) = (2*n^3 + 3*n^2 + 22*n + 24) / 24 for n even.
a(n) = (2*n^3 + 3*n^2 + 22*n + 21) / 24 for n odd.
(End)

a(0)=1 prepended by Alois P. Heinz, Dec 27 2023

A273751 Triangle of the natural numbers written by decreasing antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 6, 8, 10, 13, 9, 11, 14, 17, 21, 12, 15, 18, 22, 26, 31, 16, 19, 23, 27, 32, 37, 43, 20, 24, 28, 33, 38, 44, 50, 57, 25, 29, 34, 39, 45, 51, 58, 65, 73, 30, 35, 40, 46, 52, 59, 66, 74, 82, 91, 36, 41, 47, 53, 60, 67, 75, 83, 92, 101, 111

Offset: 1

Paul Curtz, May 30 2016

nonn

A permutation of the natural numbers.
a(n) and A091995(n) are different at the ninth term.
Antidiagonal sums: 1, 2, 7, 11, ... = A235355(n+1). Same idea.
Row sums: 1, 5, 16, 37, 72, 124, 197, 294, ... = 7*n^3/12 -n^2/8 +5*n/12 +1/16 -1/16*(-1)^n with g.f. x*(1+2*x+3*x^2+x^3) / ( (1+x)*(x-1)^4 ). The third difference is of period 2: repeat [3, 4].
Indicates the order in which electrons fill the different atomic orbitals (s,p,d,f,g,h). - Alexander Goebel, May 12 2020

			1,
2,   3,
4,   5,  7,
6,   8, 10, 13,
9,  11, 14, 17, 21,
12, 15, 18, 22, 26, 31,
16, 19, 23, 27, 32, 37, 43,
20, etc.

Cf. A002061 (right diagonal), A002620 (first column), A033638, A091995, A234305 (antidiagonals of the triangle).

A273751 := proc(n,k)
    option remember;
    if k = n then
        A002061(n) ;
    elif k > n or k < 0 then
        0;
    elif k = n-1 then
        procname(n-1,k)+k ;
    else
        procname(n-1,k+1)+1 ;
    end if;
end proc: # R. J. Mathar, Jun 13 2016

T[n_, k_] := T[n, k] = Which[k == n, n(n-1) + 1, k == n-1, (n-1)^2 + 1, k == 1, n + T[n-2, 1], 1 < k < n-1, T[n-1, k+1] + 1,True, 0];
Table[T[n, k], {n, 12}, {k, n}] // Flatten (* Jean-François Alcover, Jun 10 2016 *)

T(n, k) = (2 * (n+k)^2 + 7 + (-1)^(n-k)) / 8 - k. - Werner Schulte, Sep 27 2024

G.f.: x*y*(1 + x^4*y + x^2*(y - 1)*y + x^5*y^2 - x^3*y*(y + 2))/((1 - x)^3*(1 + x)*(1 - x*y)^3). - Stefano Spezia, Sep 28 2024

A338885 Irregular triangle read by rows in which the n-th row lists all numbers k such that there exists a diagonal lattice rectangle touching all four sides of an n X k rectangle.

Original entry on oeis.org

2, 3, 4, 5, 4, 5, 7, 6, 9, 10, 5, 7, 8, 11, 13, 7, 8, 10, 13, 16, 17, 6, 9, 11, 12, 15, 19, 21, 6, 8, 10, 11, 14, 17, 22, 25, 26, 7, 9, 10, 11, 13, 14, 16, 17, 19, 25, 29, 31, 9, 12, 13, 15, 18, 20, 21, 28, 33, 36, 37, 7, 8, 11, 12, 13, 14, 15, 17, 20, 22, 23

Offset: 2

Peter Kagey, Nov 14 2020

A diagonal lattice rectangle is a rectangle with integer coordinates and no side parallel to the x-axis.
Conjecture: The smallest number in the n-th row is A228286(n).
Conjecture: The largest number in the n-th row is A033638(n).

			Table begins:
   n | n-th row
-----+------------------------------------------------
   2 | 2
   3 | 3
   4 | 4,  5
   5 | 4,  5,  7
   6 | 6,  9, 10
   7 | 5,  7,  8, 11, 13
   8 | 7,  8, 10, 13, 16, 17
   9 | 6,  9, 11, 12, 15, 19, 21
  10 | 6,  8, 10, 11, 14, 17, 22, 25, 26
  11 | 7,  9, 10, 11, 13, 14, 16, 17, 19, 25, 29, 31
  12 | 9, 12, 13, 15, 18, 20, 21, 28, 33, 36, 37
For n = 6, three of the diagonal lattice rectangles that touch the y-axis, x-axis, and line x = 6 are:
(2 ,6), (0,2), (4,0), (6,4);
(2, 9), (0,8), (4,0), (6,1); and
(3,10), (0,9), (3,0), (6,1);
which have maximum y-values of 6, 9, and 10 respectively.

Peter Kagey, Table of n, a(n) for n = 2..11808 (first 100 rows, flattened)
Code Golf Stack Exchange, Rectangles in rectangles

Cf. A033638, A085582, A113751, A228286.

Cf. A338886 (row lengths).

A342472 T(n,k) is the maximum sum of products of adjacent parts in all compositions of n into k parts: triangle read by rows.

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 0, 4, 4, 3, 0, 6, 6, 5, 4, 0, 9, 9, 8, 6, 5, 0, 12, 12, 11, 9, 7, 6, 0, 16, 16, 15, 12, 10, 8, 7, 0, 20, 20, 19, 16, 13, 11, 9, 8, 0, 25, 25, 24, 20, 17, 14, 12, 10, 9, 0, 30, 30, 29, 25, 21, 18, 15, 13, 11, 10, 0, 36, 36, 35, 30, 26, 22, 19, 16, 14, 12, 11, 0, 42, 42, 41

Offset: 1

R. J. Mathar, Mar 13 2021

Denote compositions of n into k parts by n = p_1 +p_2 + .... +p_k, p_i>0. For these compositions let S(n,k,c) = p_1*p_2 +p_2*p_3 +.. +p_{k-1}*p_k. Then T(n,k) = max_c S(n,k,c), where c runs through all A007318(n-1,k-1) compositions.
Background: Let p_i be the number of elements in level i of a poset of n points. Connect all points on level i with all points on level i+1 "maximally" with p_i*p_{i+1} arcs in the Hasse diagram. So T(n,k) is a lower bound on the maximum number of arcs in a Hasse diagram with k levels, and the maximum T(n,k)  (+1 to add the diagrams of n disconnected elements) of a row is a lower bound of the row lengths of A342447.
T(n,2) = A002620(n) has the standard interpretation of maximizing the area p_1*p_2 of a rectangle given the semiperimeter p_1+p_2=n. [S=p_1*p_2=p_1*(n-p_1) is a quadratic function of p_1 with well defined maximum.] - R. J. Mathar, Mar 14 2021
T(n,3) maximizes S = +p_1*p_2+p_2*p_3 = p_1*p_2+p_2*(n-p_1-p_2) = p_2*(n-p_2) which again is a quadratic function of p_2 with well defined maximum. - R. J. Mathar, Mar 14 2021
For k>=4 and odd n-k consider p_1=1, p_2=(n-k+1)/2, p_3=p_2+1, p_4=p_5=..=p_k=1 which gives S= n+(n-k)+[(n-k)^2-5]/4, a lower bound (apparently strict). For k>=4 and even n-k consider p_1=1, p_2=p_3=(n-k+2)/2, p_4=p_5=...=p_k=1 which gives S=n-2+(n-k+2)^2/4, a lower bound (apparently strict). - R. J. Mathar, Mar 14 2021

			For n=6 and k=3 for example 6 = 2+3+1 = 1+3+2 obtain 2*3+3*1 = 9 = T(6,3).
For n=6 and k=4 for example 6 = 1+2+2+1 obtains 1*2+2*2+2*1=8 =T(6,4).
For n=7 and k=4 for example 7 = 1+3+2+1 = 1+2+3+1 obtains 1*2+2*3+3*1 = 11 = T(7,4).
For n=7 and k=5 for example 7 = 1+1+2+2+1 = 1+2+2+1+1 obtains 1*2+2*2+2*1+1*1 = 9 = T(7,5).
The triangle starts with n>=1 and 1<=k<=n as:
  0
  0   1
  0   2   2
  0   4   4   3
  0   6   6   5   4
  0   9   9   8   6   5
  0  12  12  11   9   7   6
  0  16  16  15  12  10   8   7
  0  20  20  19  16  13  11   9   8
  0  25  25  24  20  17  14  12  10   9
  0  30  30  29  25  21  18  15  13  11  10
  0  36  36  35  30  26  22  19  16  14  12  11
  0  42  42  41  36  31  27  23  20  17  15  13  12
  0  49  49  48  42  37  32  28  24  21  18  16  14  13
  0  56  56  55  49  43  38  33  29  25  22  19  17  15  14

Cf. A002620 (columns 2,3,5 ?), A024206 (column 4?), A033638 (column 6?), A290743 (column 7?), A342447.

# Maximum of Sum_i  p_i*p(i+1) over all combinations n=p_1+p_2+..p_k
A342472 := proc(n,k)
    local s,c;
    s := 0 ;
    for c in combinat[composition](n,k) do
        add( c[i]*c[i+1],i=1..nops(c)-1) ;
        s := max(s,%) ;
    end do:
    s ;
end proc:
for n from 1 to 15 do
    for k from 1 to n do
        printf("%3d ",A342472(n,k)) ;
    end do:
    printf("\n") ;
end do:

T(n,n) = n-1; where all p_i=1.
T(n,2) = T(n,3) = A002620(n).
T(n,k) >= 2*n-k+((n-k)^2-5)/4, n-k odd, k>=4. - R. J. Mathar, Mar 14 2021
T(n,k) >= n-2+(n-k+2)^2/4, n-k even, k>=4. - R. J. Mathar, Mar 14 2021

A375925 Squares visited by a king moving on a walled, spirally numbered board, where a wall must be jumped on each move, always to the lowest available unvisited square.

Original entry on oeis.org

1, 4, 14, 3, 11, 2, 8, 22, 7, 19, 5, 15, 33, 13, 29, 12, 28, 10, 24, 9, 23, 45, 21, 41, 20, 6, 18, 38, 17, 35, 16, 34, 60, 32, 58, 31, 55, 30, 54, 86, 52, 26, 48, 25, 47, 77, 46, 76, 44, 74, 43, 71, 42, 70, 40, 68, 39, 67, 37, 63, 36, 62, 96, 61, 95, 59, 93

Offset: 1

Sameer Khan, Sep 03 2024

nonn

The board is numbered with the following walled, square spiral:
.
  17  16  15  14  13 | .
     -------------   | .
  18 | 5   4   3 |12 | .
     |   -----   |   | .
  19 | 6 | 1   2 |11 | .
     |   ---------   | .
  20 | 7   8   9  10 | .
     ----------------- .
  21  22  23  24  25  26
.
The walls mark the boundary of the spiral.
A line drawn from the center of the starting square of a king move to the center of the ending square must pass through a wall. The king jumps over that wall. Some moves would just touch a wall without passing through the wall (e.g. 1 to 3). Such moves are not permissible.
The rules imply that the king cannot move from a square labeled k in the spiral to a square labeled k +- 1 or k +- 2.
Comment from M. F. Hasler, May 08 2025 (Start)
The sequence appears to be a permutation of the positive integers. The path drawn by Kevin Ryde shows the quasi-periodic structure of the trajectory and may lead to a formal proof.
However, it would be more natural to start the path at the origin, at a square labeled n = 0 (to which the king never moves). Then the sequence would conjecurally be a permutation of the nonnegative integers. This also leads to a more natural numbering for the squares in terms of the x,y coordinates - compare the Python function "square_number()". See A383185. (End) [Comment edited by N. J. A. Sloane, May 14 2025 following discussion with Kevin Ryde.]

			For n = 2, a(2) = 4 because moving to 2 or 3 does not pass through a wall.

Sameer Khan, Table of n, a(n) for n = 1..100
Kevin Ryde, Path Plot

Cf. A033638, A316667 (trapped knight), A336038 (trapped king).

Cf. A383185 (zero-indexed variant), A316328 (knight's path).

def square_number(z): return int(4*y**2-y-x if (y := z.imag) >= abs(x := z.real)
    else 4*x**2-x-y if -x>=abs(y) else (4*y-3)*y+x if -y>=abs(x) else (4*x-3)*x+y)
def A375925(n):
    if not hasattr(A:=A375925, 'terms'): A.terms=[1]; A.pos=0
    while len(A.terms) < n:
        s,d = min((s,d) for d in (1, 1+1j, 1j, 1j-1, -1, -1-1j, -1j, 1-1j) if
            abs((s:=1+square_number(A.pos+d))-A.terms[-1]) > 2 and s not in A.terms)
        A.terms.append(s); A.pos += d
    return A.terms[n-1] # M. F. Hasler, May 07 2025

a(n) = A383185(n-1)+1. - M. F. Hasler, May 12 2025

Entry revised by N. J. A. Sloane, May 12 2025

A377802 Triangle read by rows: T(n, k) = (2 * (n+1)^2 + 7 - (-1)^n) / 8 - k.

Original entry on oeis.org

1, 2, 1, 4, 3, 2, 6, 5, 4, 3, 9, 8, 7, 6, 5, 12, 11, 10, 9, 8, 7, 16, 15, 14, 13, 12, 11, 10, 20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31

Offset: 1

Werner Schulte, Nov 07 2024

The natural numbers, based on quarter-squares (A002620 and A033638); every natural number occurs exactly twice.

			Triangle T(n, k) for 1 <= k <= n starts:
n\ k :   1   2   3   4   5   6   7   8   9  10  11  12  13
==========================================================
   1 :   1
   2 :   2   1
   3 :   4   3   2
   4 :   6   5   4   3
   5 :   9   8   7   6   5
   6 :  12  11  10   9   8   7
   7 :  16  15  14  13  12  11  10
   8 :  20  19  18  17  16  15  14  13
   9 :  25  24  23  22  21  20  19  18  17
  10 :  30  29  28  27  26  25  24  23  22  21
  11 :  36  35  34  33  32  31  30  29  28  27  26
  12 :  42  41  40  39  38  37  36  35  34  33  32  31
  13 :  49  48  47  46  45  44  43  42  41  40  39  38  37
  etc.

A002620 (column 1), A024206 (column 2), A014616 (column 3), A004116 (column 4), A033638 (main diagonal), A290743 (1st subdiagonal).

Cf. A006003, A026741, A002061, A000290, A002378, A005563, A246694.

PARI
```
T(n,k)=(2*(n+1)^2+7-(-1)^n)/8-k
```

T(n, k) = A002620(n+1) + 1 - k.
T(2*n-1, n) = n^2 - n + 1 = A002061(n); T(2*n-2, n) = (n-1)^2 = A000290(n-1) for n > 1; T(2*n-3, n) = (n-1) * (n-2) = A002378(n-2) for n > 2; T(2*n-4, n) = (n-1) * (n-3) = A005563(n-3) for n > 3.
Row sums are (2 * n^3 + 5 * n - n * (-1)^n) / 8 = (A006003(n) + A026741(n)) / 2.
G.f.: x*y*(1 - x*y + x^2*y + x^4*y^2 - x^5*y^3 + x^6*y^3 - x^3*y*(1 + 2*y - y^2))/((1 - x)^3*(1 + x)*(1 - x*y)^3*(1 + x*y)). - Stefano Spezia, Nov 08 2024

A079644 a(n) = (n mod sqrtint(n)).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 2, 0, 1, 2, 0, 0, 1, 2, 3, 0, 1, 2, 3, 0, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 0, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6, 0, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 0, 1, 2, 3, 4, 5

Offset: 1

Benoit Cloitre, Jan 31 2003

Record values: given an m>=0, the first n for which a(n)=m is n = (m+1)^2+m = A028387(m). Also, for n>3, n is a square if and only if a(n)=0 and a(n-1)=0. - Stanislav Sykora, Aug 13 2014

Stanislav Sykora, Table of n, a(n) for n = 1..1023

Cf. A000290, A006446, A028387, A033638.

a:= proc(n) local r;
r:= isqrt(n);
if r^2 > n then r:= r-1 fi;
n mod r;
end proc:
seq(a(n),n=1..100); # Robert Israel, Aug 13 2014

A079644[n_]:=Mod[n,Floor[n^(1/2)]]; Array[A079644,200] (* Enrique Pérez Herrero, Oct 06 2011 *)
Table[Mod[n,Floor[Sqrt[n]]],{n,110}] (* Harvey P. Dale, Apr 10 2016 *)

PARI
```
a(n)=n%sqrtint(n)
```

a(A006446(n))=0; a(A033638(n))=1.
When n>0, a(A000290(n))=0; when n>1, a(A000290(n)-1)=0. - Stanislav Sykora, Aug 13 2014
a(n) = 0 if n or n+1 or 4*n+1 is a square, otherwise a(n) = a(n-1)+1. - Robert Israel, Aug 13 2014
G.f.: Sum_{r>=2} x^(r^2) * (x^r + 1) * ((r-1)*x^(r+1) - r*x^r + x)/(1 - x)^2. - Robert Israel, Aug 13 2014

Definition clarified by N. J. A. Sloane, Jan 11 2025

A113769 a(1) = 1, a(n+1) = a(n) + round(a(n)^(1/3)).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 14, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 47, 51, 55, 59, 63, 67, 71, 75, 79, 83, 87, 91, 95, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 176, 182, 188, 194, 200, 206, 212, 218, 224, 230, 236, 242, 248, 254

Offset: 1

Jonathan Vos Post, Jan 19 2006

A033638 a(0) = 1; a(1) = 1; for n > 1 a(n) = a(n-1) + round(sqrt(a(n-1))). Hence the current sequence is analogous to A033638, but with cube root instead of square root.

			a(19) = 43, so a(20) = a(19) + round(a(19)^(1/3)) = 43 + round(43^(1/3)) = 43 + round(3.50339806) = 43 + 4 = 47.
a(31) = 91, so a(32) = a(31) + round(a(31)^(1/3)) = 91 + round(4.49794145) = 91 + 4 = 95.
a(32) = 95, so a(33) = a(32) + round(a(32)^(1/3)) = 95 + round(4.56290264) = 95 + 5 = 100.
a(47) = 170, so a(48) = 170 + round(170^(1/3)) = 170 + round(5.53965826) = 176.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A033638, A113768 (flooring instead rounding).

NestList[#+Round[Surd[#,3]]&,1,60] (* Harvey P. Dale, Jan 12 2019 *)

a(1) = 1, a(n+1) = a(n) + round(a(n)^(1/3)).

A237513 T(n,k) = number of maximal horizontal rectangles that contain the Durfee square for partitions of n that consist of k nodes, 1 <= k <= n; triangular array read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 5, 1, 2, 1, 1, 1, 7, 1, 3, 1, 1, 1, 1, 10, 1, 5, 1, 2, 1, 1, 1, 13, 1, 7, 1, 3, 2, 1, 1, 1, 17, 1, 10, 1, 5, 3, 2, 1, 1, 1, 21, 1, 13, 1, 7, 6, 3, 1, 1, 1, 1, 26, 1, 17, 1, 10, 10, 5, 1, 3, 1, 1, 1, 31, 1

Offset: 1

Clark Kimberling, Feb 08 2014

Column n consists solely of 1's if and only if n = 1 or n is a prime. Column 4: A033638.

			Triangle begins:
  1
  1 ... 1
  1 ... 1 ... 1
  1 ... 1 ... 1 ... 3
  1 ... 1 ... 1 ... 3 ... 1
  1 ... 1 ... 1 ... 5 ... 1 ... 2
  1 ... 1 ... 1 ... 7 ... 1 ... 3 ... 1
  1 ... 1 ... 1 .. 10 ... 1 ... 5 ... 1 ... 2
  1 ... 1 ... 1 .. 13 ... 1 ... 7 ... 1 ... 3 ... 2
  1 ... 1 ... 1 .. 17 ... 1 .. 10 ... 1 ... 5 ... 3 ... 2
  1 ... 1 ... 1 .. 21 ... 1 .. 13 ... 1 ... 7 ... 6 ... 3 ... 1
  1 ... 1 ... 1 .. 26 ... 1 .. 17 ... 1 .. 10 .. 10 ... 5 ... 1 ... 3
The dimensions of maximal horizontal rectangles for the partitions of 6 are 1 X 6, 1 X 5, 2 X 2, 1 X 4, 2 X 3, 2 X 2, 1 X 3, 2 X 2, 2 X 2, 1 X 2, 1 X 1; the numbers of nodes are the products 6, 5, 4, 4, 6, 4, 3, 4, 4, 2, 1; counting the occurrences of 1,2,3,4,5,6 gives 1,1,1,5,1,2, which is row 6 of the array.

Clark Kimberling, Table of n, a(n) for n = 1..500

Cf. A033638, A000041, A033638.

durfeeSquare[part_] := Max[Map[Min, Transpose[{#, Range[Length[#]]}] &[part]]]; hRectangle[part_] := {#, part[[#]]} &[durfeeSquare[part]]; Map[Last[Transpose[Tally[Sort[Map[Times @@ hRectangle[#] &, IntegerPartitions[#]]]]]] &, Range[20]] (* Peter J. C. Moses, Feb 10 2014 *)

A245891 Number of labeled increasing unary-binary trees on n nodes whose breadth-first reading word avoids 213 and 321.

Original entry on oeis.org

1, 1, 3, 7, 20, 55, 157, 448

Offset: 1

Manda Riehl, Aug 19 2014

The number of labeled increasing unary-binary trees with an associated permutation avoiding 213 and 321 in the classical sense. The tree's permutation is found by recording the labels in the order in which they appear in a breadth-first search. (Note that a breadth-first search reading word is equivalent to reading the tree labels left to right by levels, starting with the root.)

In some cases, the same breadth-first search reading permutation can be found on differently shaped trees. This sequence gives the number of trees, not the number of permutations.

			When n=4, a(n)=7.  In the Links above we show the seven labeled increasing trees on four nodes whose permutation avoids 213 and 321.

Manda Riehl, The 7 trees when n = 4.

A126223 gives the number of binary trees instead of unary-binary trees. A033638 gives the number of permutations which avoid 213 and 321 that are breadth-first reading words on labeled increasing unary-binary trees.

cp's OEIS Frontend

A266464 Number of n X 2 binary arrays with rows and columns lexicographically nondecreasing and column sums nonincreasing.

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A273751 Triangle of the natural numbers written by decreasing antidiagonals.

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A338885 Irregular triangle read by rows in which the n-th row lists all numbers k such that there exists a diagonal lattice rectangle touching all four sides of an n X k rectangle.

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A342472 T(n,k) is the maximum sum of products of adjacent parts in all compositions of n into k parts: triangle read by rows.

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A375925 Squares visited by a king moving on a walled, spirally numbered board, where a wall must be jumped on each move, always to the lowest available unvisited square.

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A377802 Triangle read by rows: T(n, k) = (2 * (n+1)^2 + 7 - (-1)^n) / 8 - k.

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A079644 a(n) = (n mod sqrtint(n)).

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A113769 a(1) = 1, a(n+1) = a(n) + round(a(n)^(1/3)).

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