A047931 -id:A047931 - 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,

A047932 a(n) = floor(3n-sqrt(12n-3)).

Original entry on oeis.org

0, 1, 3, 5, 7, 9, 12, 14, 16, 19, 21, 24, 26, 29, 31, 34, 36, 39, 42, 44, 47, 49, 52, 55, 57, 60, 63, 65, 68, 71, 73, 76, 79, 81, 84, 87, 90, 92, 95, 98, 100, 103, 106, 109, 111, 114, 117, 120, 122, 125, 128, 131, 133, 136, 139, 142, 144, 147, 150, 153, 156, 158, 161

Offset: 1

Ralf W. Grosse-Kunstleve

nonn

a(n) = cumulative sum of number of new penny-penny contacts when putting pennies on a table following a spiral pattern. This is the maximum possible number of contacts.

a(n) is also the maximum number of times the minimum distance can occur among n points in the plane [Harborth].

Peter Kagey, Table of n, a(n) for n = 1..10000
K. Bezdek, M. A. Khan, Contact numbers for sphere packings, arXiv:1601.00145 [math.MG], 2016, Theorem 3.1.
Peter Brass, The maximum number of second smallest distances in finite planar sets, Discrete & Computational Geometry 7.1 (1992): 371-379.
R. W. Grosse-Kunstleve, Penny Spiral Sequence
H. Harborth, Solution to problem 644A, Elemente der Mathematik (EMS Publishing House) 29, 14-15.
MathOverflow, Maximal number of edges and triangular cells for n points in a triangular lattice, August 2011.

Partial sums of A047931.
A186705 is the maximum number of times the *same* distance can occur between n points in the plane, not necessarily the *minimum*.
Cf. A293956.

Table[Floor[3n-Sqrt[12n-3]],{n,70}] (* Harvey P. Dale, Dec 25 2014 *)

a(n) = floor(3*n-sqrt(12*n-3)).

Entry revised by N. J. A. Sloane, Nov 01 2017

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)

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 *)

A278181 Hexagonal spiral constructed on the nodes of the triangular net in which each new term is the sum of its neighbors.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 7, 8, 9, 12, 14, 19, 22, 29, 33, 42, 47, 59, 74, 82, 99, 108, 129, 155, 169, 202, 243, 265, 316, 378, 411, 486, 575, 622, 728, 861, 1017, 1099, 1280, 1487, 1595, 1832, 2116, 2440, 2609, 2980, 3425, 3933, 4198, 4779, 5473, 6262, 6673, 7570, 8631, 9828, 10450, 11800, 13389, 15267, 17383

Offset: 0

Omar E. Pol, Nov 14 2016

nonn

To evaluate a(n) consider the neighbors of a(n) that are present in the spiral when a(n) should be a new term in the spiral.

			Illustration of initial terms as a spiral:
.
.             22 - 19 - 14
.             /          \
.           29    3 - 2   12
.           /    /     \   \
.         33    4   1 - 1   9
.           \    \         /
.           42    5 - 7 - 8
.             \
.             47 - 59 - 74
.
a(16) = 47 because the sum of its two neighbors is 42 + 5 = 47.
a(17) = 59 because the sum of its three neighbors is 47 + 5 + 7 = 59.
a(18) = 74 because the sum of its three neighbors is 59 + 7 + 8 = 74.
a(19) = 82 because the sum of its two neighbors is 74 + 8 = 82.

JungHwan Min, Table of n, a(n) for n = 0..10000

Cf. A047931, A064642, A122479, A141481, A274821, A274921, A275606, A275610.

A278181[0] = A278181[1] = 1; A278181[n_] := A278181[n] = With[{lev = Ceiling[1/6 (-3 + Sqrt[3] Sqrt[3 + 4 n])]}, With[{pos = 3 lev (lev - 1) + (n - 3 lev (1 + lev))/lev*(lev - 1)}, A278181[n - 1] + A278181[Ceiling[pos]] + If[Mod[n, lev] == 0 || n - 3 lev (lev - 1) == 1, 0, A278181[Floor[pos]]] + If[3 lev (1 + lev) == n, A278181[n - 6 lev + 1], 0]]]; Array[A278181, 61, 0] (* JungHwan Min, Nov 21 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 *)

cp's OEIS Frontend

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

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A047932 a(n) = floor(3*n-sqrt(12*n-3)).

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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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A278181 Hexagonal spiral constructed on the nodes of the triangular net in which each new term is the sum of its neighbors.

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

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A047932 a(n) = floor(3n-sqrt(12n-3)).