A253028 A fractal array resembling the shape of a conifer tree read by rows. A mirror symmetric array of numbers where the n-th term is equal to the number of terms in the n-th row of the array.

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

Offset

1,2

Comments

A layer of the array is defined as those terms having the same distance from left and right boundary of the array. If the terms of a layer are read by rows, one obtains the sequence of positive integers 1, 2, 3, 4, 5, 6 ....

The n-th row of the array consists of a(n) terms.

The following illustration shows the first 31 rows of the array.

.......................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,

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

E. Angelini, A Xmas fractal tree, Seqfan (Dec 27 2014)

Rémy Sigrist, PARI program

Example

Start with the tip of the array consisting of three consecutive positive integers beginning with 1:
...1,
..2,3,
Then the third row of the array must consist of three terms. The outermost terms of third row belong to the first layer and are set to the next consecutive integers after 3. After that, the remaining term marked X in third row is the first term of the second layer and its value is set to 1.
...1,......1,......1,
..2,3,....2,3,....2,3,
.X,X,X,..4,X,5,..4,1,5,
Since a(4) = 4, the fourth row has four terms. Set terms of fourth row which belong to first layer to the next consecutive integers in that layer. After that, the two remaining terms in fourth row belong to second layer, so set them to next two consecutive integers after 1.
....1,........1,........1,
...2,3,......2,3,......2,3,
..4,1,5,....4,1,5,....4,1,5,
.X,X,X,X,..6,X,X,7,..6,2,3,7,
The next row has one term, since a(5) = 1. Set value of X to next integer not yet in first layer.
....1,........1,
...2,3,......2,3,
..4,1,5,....4,1,5,
.6,2,3,7,..6,2,3,7,
....X,........8,
The sixth row has five terms, since a(6) = 5. The next consecutive integers in first layer are 9 and 10. After that, the next consecutive integers in second layer are 4 and 5. The last remaining term marked X belongs to third layer, of which no terms are present in the array yet, so set its value to 1.
.....1,..........1,...........1,...........1,
....2,3,........2,3,.........2,3,.........2,3,
...4,1,5,......4,1,5,.......4,1,5,.......4,1,5,
..6,2,3,7,....6,2,3,7,.....6,2,3,7,.....6,2,3,7,
.....8,..........8,...........8,...........8,
.X,X,X,X,X,..9,X,X,X,10,..9,4,X,5,10,..9,4,1,5,10,

Prog

(PARI) See Links section.

Crossrefs

Cf. A253146, A334081.

Sequence in context: A332266 A129709 A253146 * A133108 A055441 A104717

Adjacent sequences: A253025 A253026 A253027 * A253029 A253030 A253031

Keyword

nonn,tabf

Author

Felix Fröhlich, Mar 23 2015; originally suggested by Eric Angelini, Dec 27 2014

Status

approved