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

1,2

Comments

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

Links

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Éric Angelini, A fractal tree, SeqFan list, Dec 27 2014.

Example

.   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.
From N. J. A. Sloane, Jan 04 2015: (Start)
Eric Angelini's original posting to the Sequence Fans mailing list gave a different sequence, as follows:
..................................1,
.................................2,3,
................................4,1,5,
...............................6,2,3,7,
..................................8,
..............................9,4,1,5,10,
............................11,6,2,3,7,12,
................................13,14,
...............................15,8,16,
...........................17,9,4,1,5,10,18,
.........................19,11,6,2,3,7,12,20,
........................21,13,8,4,1,5,9,14,22,
.............................23,15,16,24,
.................................25,
...........................26,17,10,18,27,
......................28,19,11,6,2,3,7,12,20,29,
.....................30,21,13,8,4,1,5,9,14,22,31,
..........................32,23,15,16,24,33,
................................34,35,
..............................36,25,37,
........................38,26,17,10,18,27,39,
...................40,28,19,11,6,2,3,7,12,20,29,41,
..................42,30,21,13,8,4,1,5,9,14,22,31,43,
................44,32,23,15,10,6,2,3,7,11,16,24,33,45,
...............46,34,25,17,12,8,4,1,5,9,13,18,26,35,47,
.......................48,36,27,19,20,28,37,49,
.............50,38,29,21,14,10,6,2,3,7,11,15,22,30,39,51,
............52,40,31,23,16,12,8,4,1,5,9,13,17,24,32,41,53,
.....................54,42,33,25,18,26,34,43,55,
.............................56,44,45,57,
.................................58,
.....................................
The idea is that the n-th term is equal to the number of terms in the n-th row of the tree. This lovely sequence (whose precise definition is not clear to me) is not yet in the OEIS. (End)
The sequence referred to is A253028. - Felix Fröhlich, May 23 2016

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 *)

Prog

(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]

Crossrefs

Cf. A253028. Row sums appear to be A035608.

Cf. A000290, A002061.

Sequence in context: A243713 A332266 A129709 * A253028 A133108 A055441

Adjacent sequences: A253143 A253144 A253145 * A253147 A253148 A253149

Keyword

nonn,tabl

Author

Eric Angelini and Reinhard Zumkeller, Dec 27 2014

Status

approved