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).

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

Author

Eric Angelini and Reinhard Zumkeller, Dec 27 2014

Keywords

• nonn
• tabl

Comments

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

Examples

			.   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.
		

Links

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

Crossrefs

Cf. A253028. Row sums appear to be A035608.

Cf. A000290, A002061.

Programs

Haskell

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]

Mathematica

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 *)