A278317 -id:A278317 - cp's OEIS frontend

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

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)

A275015 Number of neighbors of each new term in an isosceles triangle read by rows.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Nov 20 2016

To evaluate a(n) consider only the neighbors of a(n) that are present in the isosceles triangle when a(n) should be a new term in the triangle.
Apart from the left border and the right border, the rest of the elements are 3's.
If every "3" is replaced with a "4", we have the sequence A278290.
a(n) is also the number of new penny-penny contacts when putting pennies in a triangular arrangement.
For the same idea but for a right triangle see A278317; for a square array see A278290, for a square spiral see A278354; and for a hexagonal spiral see A047931.

			The sequence written as an isosceles triangle begins:
.
.                     0;
.                   1,  2;
.                 1,  3,  2;
.               1,  3,  3,  2;
.             1,  3,  3,  3,  2;
.           1,  3,  3,  3,  3,  2;
.         1,  3,  3,  3,  3,  3,  2;
.       1,  3,  3,  3,  3,  3,  3,  2;
.     1,  3,  3,  3,  3,  3,  3,  3,  2;
.   1,  3,  3,  3,  3,  3,  3,  3,  3,  2;
...

Row sums give A008585.
Left border gives A057427.
Every diagonal that is parallel to the left border gives the elements greater than 1 of A158799.
Right border gives 0 together with A007395, also twice A057427.
Every diagonal that is parallel to the right border gives A122553.
Cf. A047931, A278290, A278317, A278354.

Table[Boole[n > 1] (Prepend[Reverse@ Rest@ #, First@ #] &@ Range@ n /. k_ /; k > 3 -> 3), {n, 13}] // Flatten (* or *)
Table[Boole[n > 1] (Map[Mod[#, n] &, Range@ n] /. {k_ /; k > 1 -> 3, 0 -> 2}), {n, 13}] // Flatten (* Michael De Vlieger, Nov 23 2016 *)

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 *)

A278480 Number of neighbors of the n-th term in a full right triangle read by rows.

Original entry on oeis.org

2, 4, 5, 5, 7, 5, 5, 8, 7, 5, 5, 8, 8, 7, 5, 5, 8, 8, 8, 7, 5, 5, 8, 8, 8, 8, 7, 5, 5, 8, 8, 8, 8, 8, 7, 5, 5, 8, 8, 8, 8, 8, 8, 7, 5, 5, 8, 8, 8, 8, 8, 8, 8, 7, 5, 5, 8, 8, 8, 8, 8, 8, 8, 8, 7, 5, 5, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 5, 5, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 5, 5, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 5

Offset: 1

Omar E. Pol, Nov 23 2016

Apart from the first column and the first two diagonals, the rest of the elements are 8's.

For the same idea but for an isosceles triangle see A278481; for a square array see A278545, for a square spiral see A010731; and for a hexagonal spiral see A010722.

			The sequence written as a right triangle begins:
2;
4, 5;
5, 7, 5;
5, 8, 7, 5;
5, 8, 8, 7, 5;
5, 8, 8, 8, 7, 5;
5, 8, 8, 8, 8, 7, 5;
5, 8, 8, 8, 8, 8, 7, 5;
5, 8, 8, 8, 8, 8, 8, 7, 5;
5, 8, 8, 8, 8, 8, 8, 8, 7, 5;
...

Row sums give 2 together with the elements > 1 of A017077.
Also, row sums give 2 together with A004768.
Cf. A010722, A010731, A278317, A278481, A278545.

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.

A278545 Number of neighbors of the n-th term in a full square array read by antidiagonals.

Original entry on oeis.org

3, 5, 5, 5, 8, 5, 5, 8, 8, 5, 5, 8, 8, 8, 5, 5, 8, 8, 8, 8, 5, 5, 8, 8, 8, 8, 8, 5, 5, 8, 8, 8, 8, 8, 8, 5, 5, 8, 8, 8, 8, 8, 8, 8, 5, 5, 8, 8, 8, 8, 8, 8, 8, 8, 5, 5, 8, 8, 8, 8, 8, 8, 8, 8, 8, 5, 5, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 5, 5, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 5, 5, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 5

Offset: 1

Omar E. Pol, Nov 23 2016

Apart from the first row and the first column, the rest of the elements are 8's.

For the same idea but for a right triangle see A278480; for an isosceles triangle see A278481; for a square spiral see A010731; and for a hexagonal spiral see A010722.

			The corner of the square array begins:
3,5,5,5,5,5,5,5,5,5,...
5,8,8,8,8,8,8,8,8,...
5,8,8,8,8,8,8,8,...
5,8,8,8,8,8,8,...
5,8,8,8,8,8,...
5,8,8,8,8,...
5,8,8,8,...
5,8,8,...
5,8,...
5,...
...

Robert Israel, Table of n, a(n) for n = 1..10000

Antidiagonal sums give 3 together with the elements > 2 of A017089.

Cf. A010722, A010731, A274912, A274913, A278317, A278290, A278480, A278481.

3, seq(op([5,8$i,5]),i=0..20); # Robert Israel, Dec 04 2016

G.f. 3+x+8*x/(1-x)-3*(1+x)*Theta_2(0,sqrt(x))/(2*x^(1/8)) where Theta_2 is a Jacobi Theta function. - Robert Israel, Dec 04 2016

A278645 Triangle read by rows in which each new term is the sum of its two largest neighbors in the structure.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 8, 15, 22, 29, 23, 45, 74, 103, 132, 68, 142, 245, 377, 509, 641, 210, 455, 832, 1341, 1982, 2623, 3264, 665, 1497, 2838, 4820, 7443, 10707, 13971, 17235, 2162, 5000, 9820, 17263, 27970, 41941, 59176, 76411, 93646, 7162, 16982, 34245, 62215, 104156, 163332, 239743, 333389, 427035, 520681

Offset: 1

Omar E. Pol, Nov 24 2016

To evaluate T(n,k) consider only the two largest neighbors of T(n,k) that are present in the triangle when T(n,k) should be a new term in the triangle.
For the same idea but for a square spiral see A278180; and for a hexagonal spiral see A278619.
It appears that the same idea for an isosceles triangle and also for a square array gives A030237.

			Triangle begins:
1;
1,    2;
3,    5,     7;
8,    15,    22,    29;
23,   45,    74,    103,   132;
68,   142,   245,   377,   509,    641;
210,  455,   832,   1341,  1982,   2623,   3264;
665,  1497,  2838,  4820,  7443,   10707,  13971,  17235;
2162, 5000,  9820,  17263, 27970,  41941,  59176,  76411,  93646;
7162, 16982, 34245, 62215, 104156, 163332, 239743, 333389, 427035, 520681;
...

Cf. A030237, A278180, A278317, A278619.

cp's OEIS Frontend

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

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A275015 Number of neighbors of each new term in an isosceles triangle read by rows.

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

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A278480 Number of neighbors of the n-th term in a full right triangle read by rows.

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

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A278545 Number of neighbors of the n-th term in a full square array read by antidiagonals.

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A278645 Triangle read by rows in which each new term is the sum of its two largest neighbors in the structure.

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