A275015 -id:A275015 - cp's OEIS frontend

A278317 Number of neighbors of each new term in a right triangle read by rows.

0, 1, 2, 2, 3, 2, 2, 4, 3, 2, 2, 4, 4, 3, 2, 2, 4, 4, 4, 3, 2, 2, 4, 4, 4, 4, 3, 2, 2, 4, 4, 4, 4, 4, 3, 2, 2, 4, 4, 4, 4, 4, 4, 3, 2, 2, 4, 4, 4, 4, 4, 4, 4, 3, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 3, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 2

Offset: 1

Omar E. Pol, Nov 18 2016

To evaluate T(n,k) consider only the neighbors of T(n,k) that are present in the triangle when T(n,k) should be a new term in the triangle.
Apart from the first column and the first two diagonals the rest of the elements are 4's.
For the same idea but for an isosceles triangle see A275015; for a square array see A278290, for a square spiral see A278354; and for a hexagonal spiral see A047931.

			Triangle begins:
0;
1, 2;
2, 3, 2;
2, 4, 3, 2;
2, 4, 4, 3, 2;
2, 4, 4, 4, 3, 2;
2, 4, 4, 4, 4, 3, 2;
2, 4, 4, 4, 4, 4, 3, 2;
2, 4, 4, 4, 4, 4, 4, 3, 2;
2, 4, 4, 4, 4, 4, 4, 4, 3, 2;
...

Apart from the initial zero, row sums give A004767.
Column 1 is A130130.
Columns > 1 give the terms greater than 1 of A158411.
Right border gives 0 together with A007395, also twice A057427.
Second right border gives A122553.
Cf. A047931, A274912, A274913, A275015, A278290, A278317, A278354,

A278354 Number of neighbors of each new term in a square spiral.

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 2, 4, 3, 2, 4, 3, 2, 4, 4, 3, 2, 4, 4, 3, 2, 4, 4, 4, 3, 2, 4, 4, 4, 3, 2, 4, 4, 4, 4, 3, 2, 4, 4, 4, 4, 3, 2, 4, 4, 4, 4, 4, 3, 2, 4, 4, 4, 4, 4, 3, 2, 4, 4, 4, 4, 4, 4, 3, 2, 4, 4, 4, 4, 4, 4, 3, 2, 4, 4, 4, 4, 4, 4, 4, 3, 2, 4, 4, 4, 4, 4, 4, 4, 3, 2, 4, 4, 4, 4, 4, 4, 4, 4, 3, 2, 4, 4, 4, 4

Offset: 1

Omar E. Pol, Nov 19 2016

nonn

Here the "neighbors" of a(n) are defined to be the adjacent elements to a(n) in the same row, column or diagonals, that are present in the spiral when a(n) is the new element of the sequence in progress.

For the same idea but for a right triangle see A278317; for an isosceles triangle see A275015; for a square array see A278290; and for a hexagonal spiral see A047931.

			Illustration of initial terms as a spiral (n = 1..169):
.
.     2 - 3 - 4 - 4 - 4 - 4 - 4 - 4 - 4 - 4 - 4 - 4 - 2
.     |                                               |
.     4   2 - 3 - 4 - 4 - 4 - 4 - 4 - 4 - 4 - 4 - 2   3
.     |   |                                       |   |
.     4   4   2 - 3 - 4 - 4 - 4 - 4 - 4 - 4 - 2   3   4
.     |   |   |                               |   |   |
.     4   4   4   2 - 3 - 4 - 4 - 4 - 4 - 2   3   4   4
.     |   |   |   |                       |   |   |   |
.     4   4   4   4   2 - 3 - 4 - 4 - 2   3   4   4   4
.     |   |   |   |   |               |   |   |   |   |
.     4   4   4   4   4   2 - 3 - 2   3   4   4   4   4
.     |   |   |   |   |   |       |   |   |   |   |   |
.     4   4   4   4   4   3   0 - 1   4   4   4   4   4
.     |   |   |   |   |   |           |   |   |   |   |
.     4   4   4   4   3   2 - 4 - 3 - 2   4   4   4   4
.     |   |   |   |   |                   |   |   |   |
.     4   4   4   3   2 - 4 - 4 - 4 - 3 - 2   4   4   4
.     |   |   |   |                           |   |   |
.     4   4   3   2 - 4 - 4 - 4 - 4 - 4 - 3 - 2   4   4
.     |   |   |                                   |   |
.     4   3   2 - 4 - 4 - 4 - 4 - 4 - 4 - 4 - 3 - 2   4
.     |   |                                           |
.     3   2 - 4 - 4 - 4 - 4 - 4 - 4 - 4 - 4 - 4 - 3 - 2
.     |
.     2 - 4 - 4 - 4 - 4 - 4 - 4 - 4 - 4 - 4 - 4 - 4 - 3
.

Cf. A002620, A033638, A047931, A141481, A274917, A275015, A275609, A278180, A278290, A278317.

0,1,seq(op([2,4$floor(i/2),3]),i=0..30); # Robert Israel, Nov 22 2016

From Robert Israel, Nov 22 2016: (Start)
a(n) = 3 if n>=4 is in A002620.
a(n) = 2 if n>=2 is in A033638.
Otherwise, a(n) = 4 if n > 2. (End)

A278290 Number of neighbors of each new term in a square array read by antidiagonals.

Original entry on oeis.org

0, 1, 2, 1, 4, 2, 1, 4, 4, 2, 1, 4, 4, 4, 2, 1, 4, 4, 4, 4, 2, 1, 4, 4, 4, 4, 4, 2, 1, 4, 4, 4, 4, 4, 4, 2, 1, 4, 4, 4, 4, 4, 4, 4, 2, 1, 4, 4, 4, 4, 4, 4, 4, 4, 2, 1, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 1, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 1, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 1, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 1, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2

Offset: 1

Omar E. Pol, Nov 16 2016

Here the "neighbors" of T(n,k) are defined to be the adjacent elements to T(n,k), in the same row, column or diagonals, that are present in the square array when T(n,k) is the new element of the sequence in progress.
Apart from row 1 and column 1 the rest of the elements are 4's.
If every "4" is replaced with a "3" we have the sequence A275015.
For the same idea but for a right triangle see A278317; for an isosceles triangle see A275015; for a square spiral see A278354; and for a hexagonal spiral see A047931.

			The corner of the square array read by antidiagonals upwards begins:
0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,...
1, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4,...
1, 4, 4, 4, 4, 4, 4, 4, 4, 4,...
1, 4, 4, 4, 4, 4, 4, 4, 4,...
1, 4, 4, 4, 4, 4, 4, 4,...
1, 4, 4, 4, 4, 4, 4,...
1, 4, 4, 4, 4, 4,...
1, 4, 4, 4, 4,...
1, 4, 4, 4,...
1, 4, 4,...
1, 4,...
1,...
..

Antidiagonal sums give 0 together with A004767.
Row 1 gives 0 together with A007395, also twice A057427.
Column 1 gives A057427.
Cf. A047931, A274912, A274913, A275015, A278317, A278354.

Table[Boole[# > 1] + 2 Boole[k > 1] + Boole[And[# > 1, k > 1]] &[n - k + 1], {n, 14}, {k, n}] // Flatten (* or *)
Table[Boole[n > 1] (Map[Mod[#, n] &, Range@ n] /. {k_ /; k > 1 -> 4, 0 -> 2}), {n, 14}] // Flatten (* Michael De Vlieger, Nov 23 2016 *)

A278481 Number of neighbors of the n-th term in a full isosceles triangle read by rows.

Original entry on oeis.org

2, 4, 4, 4, 6, 4, 4, 6, 6, 4, 4, 6, 6, 6, 4, 4, 6, 6, 6, 6, 4, 4, 6, 6, 6, 6, 6, 4, 4, 6, 6, 6, 6, 6, 6, 4, 4, 6, 6, 6, 6, 6, 6, 6, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 4

Offset: 1

Omar E. Pol, Nov 23 2016

Apart from the left border and the right border, the rest of the elements are 6's.

For the same idea but for a right triangle see A278480; for a square array see A278545, for a square spiral see A010731; and for a hexagonal spiral see A010722.

			The sequence written as an isosceles triangle begins:
.
.                     2;
.                   4,  4;
.                 4,  6,  4;
.               4,  6,  6,  4;
.             4,  6,  6,  6,  4;
.           4,  6,  6,  6,  6,  4;
.         4,  6,  6,  6,  6,  6,  4;
.       4,  6,  6,  6,  6,  6,  6,  4;
.     4,  6,  6,  6,  6,  6,  6,  6,  4;
.   4,  6,  6,  6,  6,  6,  6,  6,  6,  4;
...

Row sums give A016933.
Left border gives A040002, the same as the right border.
Middle column gives the elements > 1 of A134201, also twice A122553.
Cf. A010722, A010731, A275015, A278317, A278480, A278545.

cp's OEIS Frontend

A278317 Number of neighbors of each new term in a right triangle read by rows.

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A278354 Number of neighbors of each new term in a square spiral.

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A278290 Number of neighbors of each new term in a square array read by antidiagonals.

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Mathematica

A278481 Number of neighbors of the n-th term in a full isosceles triangle read by rows.

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