A253028 -id:A253028 - cp's OEIS frontend

A354576 Variant of A253028 using only odd numbers: a mirror symmetric array of odd numbers where the n-th term is equal to the number of terms in the n-th row of the array.

1, 3, 1, 5, 7, 9, 3, 1, 5, 11, 13, 7, 3, 1, 5, 9, 15, 17, 11, 7, 3, 1, 5, 9, 13, 19, 21, 15, 23, 25, 27, 17, 11, 19, 29, 31, 21, 13, 7, 3, 1, 5, 9, 15, 23, 33, 35, 25, 17, 11, 7, 3, 1, 5, 9, 13, 19, 27, 37, 39, 29, 21, 15, 23, 31, 41

Offset: 1

Felix Fröhlich, May 30 2022

See the examples in A253028 for a detailed illustration of how the array is constructed.

			1.......................1
2.....................3,1,5
3.......................7
4...................9,3,1,5,11
5................13,7,3,1,5,9,15
6.............17,11,7,3,1,5,9,13,19
7................... 21,15,23
8...................... 25
9.................27,17,11,19,29
10.........31,21,13,7,3,1,5,9,15,23,33
11......35,25,17,11,7,3,1,5,9,13,19,27,37
12.............39,29,21,15,23,31,41

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program

Cf. A253028, A354577.

PARI
```
See Links section.
```

A354577 Variant of A253028 using only even numbers: a mirror symmetric array of even numbers where the n-th term is equal to the number of terms in the n-th row of the array.

Original entry on oeis.org

2, 4, 6, 2, 4, 8, 10, 6, 2, 4, 8, 12, 14, 16, 18, 10, 12, 20, 22, 14, 6, 2, 4, 8, 16, 24, 26, 18, 10, 6, 2, 4, 8, 12, 20, 28, 30, 22, 14, 16, 24, 32, 34, 36, 38, 26, 28, 40, 42, 30, 18, 10, 12, 20, 32, 44, 46, 34, 22, 14, 6, 2, 4, 8, 16, 24, 36, 48, 50, 38, 26

Offset: 1

Felix Fröhlich, May 30 2022

See the examples in A253028 for a detailed illustration of how the array is constructed.

			1.........................2,4
2.......................6,2,4,8
3....................10,6,2,4,8,12
4........................14,16
5.....................18,10,12,20
6.................22,14,6,2,4,8,16,24
7..............26,18,10,6,2,4,8,12,20,28
8..................30,22,14,16,24,32
9........................34,36
10....................38,26,28,40
11..............42,30,18,10,12,20,32,44
12..........46,34,22,14,6,2,4,8,16,24,36,48
13.......50,38,26,18,10,6,2,4,8,12,20,28,40,52

Rémy Sigrist, PARI program

Cf. A253028, A354576.

PARI
```
See Links section.
```

A355091 Variant of A253028 using only prime numbers.

Original entry on oeis.org

2, 3, 5, 2, 7, 11, 3, 2, 5, 13, 17, 19, 23, 7, 3, 2, 5, 11, 29, 31, 13, 7, 3, 2, 2, 3, 5, 11, 17, 37, 41, 19, 43, 47, 53, 59, 23, 13, 29, 61, 67, 31, 17, 7, 5, 3, 2, 5, 7, 11, 19, 37, 71, 73, 41, 23, 13, 11, 7, 3, 2, 2, 3, 5, 11, 13, 17, 29, 43, 79

Offset: 1

Rémy Sigrist, Jun 19 2022

See the examples in A253028 for a detailed illustration of how the array is constructed:
- the n-th term gives the length of the n-th row.
- if you remove the eventual leading and trailing term in each row, you get the prime numbers, in natural order,
- repeating this procedure with the remaining terms always yields the prime numbers, in natural order.

			The first terms / rows are:
  n   a(n)  n-th row
  --  ----  ------------------------------------------------------------
   1     2                               2, 3
   2     3                             5, 2, 7
   3     5                         11, 3, 2, 5, 13
   4     2                              17, 19
   5     7                     23, 7, 3, 2, 5, 11, 29
   6    11              31, 13, 7, 3, 2, 2, 3, 5, 11, 17, 37
   7     3                            41, 19, 43
   8     2                              47, 53
   9     5                        59, 23, 13, 29, 61
  10    13          67, 31, 17, 7, 5, 3, 2, 5, 7, 11, 19, 37, 71
  11    17  73, 41, 23, 13, 11, 7, 3, 2, 2, 3, 5, 11, 13, 17, 29, 43, 79

Rémy Sigrist, Table of n, a(n) for n = 1..10022
Rémy Sigrist, PARI program

Cf. A253028, A354576, A354577.

PARI
```
See Links section.
```

A253146 A fractal tree, read by rows: for n > 2, T(n,1) = T(n-1,1)+2, T(n,n) = T(n-1,1)+3, and for k=2..n-1, T(n,k) = T(n-2,k-1).

Original entry on oeis.org

1, 2, 3, 4, 1, 5, 6, 2, 3, 7, 8, 4, 1, 5, 9, 10, 6, 2, 3, 7, 11, 12, 8, 4, 1, 5, 9, 13, 14, 10, 6, 2, 3, 7, 11, 15, 16, 12, 8, 4, 1, 5, 9, 13, 17, 18, 14, 10, 6, 2, 3, 7, 11, 15, 19, 20, 16, 12, 8, 4, 1, 5, 9, 13, 17, 21, 22, 18, 14, 10, 6, 2, 3, 7, 11, 15, 19, 23

Offset: 1

Eric Angelini and Reinhard Zumkeller, Dec 27 2014

Eric Angelini's original posting to the Sequence Fans mailing list gave a similar but different lovely sequence, which is now A253028. - N. J. A. Sloane, Jan 04 2015, and Felix Fröhlich, May 23 2016
It appears that:
1) partial sums of terms, situated on the outer leftmost leftwise triangle diagonal are equal to A002061(k), k>=1;
2) partial sums of terms, situated on the second (from the left) leftwise triangle diagonal represent recurrence a(k+1) = ((k-1)*a(k))/(k-3)-(2*(k+3))/(k-3), k>=3
3) partial sums of terms, situated on the outer rightmost rightwise triangle diagonal are equal to A000290(k)=k^2, k>=1. - Alexander R. Povolotsky, Dec 28 2014

			.   1:                         1
.   2:                       2   3
.   3:                     4   1   5
.   4:                   6   2   3   7
.   5:                 8   4   1   5   9
.   6:              10   6   2   3   7  11
.   7:            12   8   4   1   5   9  13
.   8:          14  10   6   2   3   7  11  15
.   9:        16  12   8   4   1   5   9  13  17
.  10:      18  14  10   6   2   3   7  11  15  19
.  11:    20  16  12   8   4   1   5   9  13  17  21
.  12:  22  18  14  10   6   2   3   7  11  15  19  23 .
Removing the first and last entries from each row gives the same tree back again.

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
Éric Angelini, A fractal tree, SeqFan list, Dec 27 2014.

Cf. A253028. Row sums appear to be A035608.

Cf. A000290, A002061.

a253146 n k = a253146_tabl !! (n-1) !! (k-1)
a253146_row n = a253146_tabl !! (n-1)
a253146_tabl = [1] : [2,3] : f [1] [2,3] where
   f us vs@(v:_) = ws : f vs ws where
                   ws = [v + 2] ++ us ++ [v + 3]

T[n_, 1] := 2n - 2;
T[n_, n_] := 2n - 1;
T[n_, k_] := T[n, k] = T[n-2, k-1];
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 20 2021 *)

A367251 Lexicographically earliest sequence starting 1,2 which can be arranged in a mirror symmetric array shape such that a(n) is the length of the n-th row and no column has the same value more than once.

Original entry on oeis.org

1, 2, 1, 2, 1, 2, 3, 3, 3, 1, 4, 1, 2, 5, 2, 3, 6, 3, 7, 1, 4, 4, 1, 8, 5, 5, 1, 4, 9, 4, 1, 6, 6, 5, 10, 5, 1, 2, 7, 7, 2, 1, 6, 11, 6, 1, 2, 7, 12, 7, 2, 1, 13, 3, 8, 8, 3, 4, 9, 9, 4, 14, 1, 2, 5, 10, 10, 5, 2, 1, 3, 8, 15, 8, 3, 4, 9, 16, 9, 4, 17, 6, 11, 11, 6, 1, 2, 5, 10, 18, 10

Offset: 1

Neal Gersh Tolunsky, Nov 11 2023

For row 5 onward, the row contents are mirror symmetric too (palindromes), as well as the shape.

Terms in the same column are successive positive integers (with some initial exceptions before row 5).

			Array (or "tree") begins, with mirror symmetry in row 5 and beyond:
  columns   v  v  v  v  v  v  v
  row 1:             1,
  row 2:          2,    1,
  row 3:             2,
  row 4:          1,    2,
  row 5:             3,
  row 6:          3,    3,
  row 7:       1,    4,    1,
  row 8:       2,    5,    2,
  row 9:       3,    6,    3,
  row 10:            7,
  row 11:   1,    4,    4,    1,
  row 12:            8,
  row 13:         5,    5,

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000
Thomas Scheuerle, blue: scatter plot of a(1) to a(10000); red: length of the row where a(n) is contained.
Neal Gersh Tolunsky, First differences of first 100000 terms.
Neal Gersh Tolunsky, Ordinal transform of first 100000 terms.
Neal Gersh Tolunsky, Graph of first 100000 terms.

Cf. A334081, A253028.

function a = A367251( max_n )
    a = [1 2 1 2 1 2];
    odd = zeros(1,max_n); even = odd;
    odd(1) = 2; even(1)= 2; c = 5;
    while  length(a) < max_n
        if mod(a(c),2) == 1
            odd(1:(a(c)+1)/2) = odd(1:(a(c)+1)/2)+1;
            a = [a odd((a(c)+1)/2:-1:2) odd(1:(a(c)+1)/2)];
        else
            even(1:a(c)/2) = even(1:a(c)/2)+1;
            a = [a even(a(c)/2:-1:1) even(1:a(c)/2)];
        end
        c = c + 1;
    end
end % Thomas Scheuerle, Nov 21 2023

cp's OEIS Frontend

A354576 Variant of A253028 using only odd numbers: a mirror symmetric array of odd numbers where the n-th term is equal to the number of terms in the n-th row of the array.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A354577 Variant of A253028 using only even numbers: a mirror symmetric array of even numbers where the n-th term is equal to the number of terms in the n-th row of the array.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A355091 Variant of A253028 using only prime numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A253146 A fractal tree, read by rows: for n > 2, T(n,1) = T(n-1,1)+2, T(n,n) = T(n-1,1)+3, and for k=2..n-1, T(n,k) = T(n-2,k-1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

A367251 Lexicographically earliest sequence starting 1,2 which can be arranged in a mirror symmetric array shape such that a(n) is the length of the n-th row and no column has the same value more than once.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

MATLAB