A278180 -id:A278180 - cp's OEIS frontend

A334742 Pascal's spiral: starting with a(1) = 1, proceed in a square spiral, computing each term as the sum of horizontally and vertically adjacent prior terms.

1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 10, 12, 12, 14, 17, 20, 20, 23, 27, 32, 37, 37, 42, 48, 55, 62, 62, 69, 77, 87, 99, 111, 111, 123, 137, 154, 174, 194, 194, 214, 237, 264, 296, 333, 370, 370, 407, 449, 497, 552, 614, 676, 676, 738, 807, 884, 971, 1070

Offset: 1

Alec Jones and Peter Kagey, May 09 2020

This is the square spiral analogy of Pascal's triangle thought of as a table read by antidiagonals.

			Spiral begins:
  111--99--87--77--69--62
                        |
   12--12--10---8---7  62
    |               |   |
   14   2---2---1   7  55
    |   |       |   |   |
   17   3   1---1   6  48
    |   |           |   |
   20   3---4---5---5  42
    |                   |
   20--23--27--32--37--37
a(15) = 10 = 8 + 2, the sum of the cells immediately to the right and below. The term to the left is not included in the sum because it has not yet occurred in the spiral.

Peter Kagey, Table of n, a(n) for n = 1..10000
Peter Kagey, Bitmap illustrating the parity of the first 2^20=1048576 terms. (Even and odd numbers are represented with black and white pixels respectively.)

Cf. A007318, A141481, A334688.
Cf. A180714, A214526, A274640, A278180, A304586, A307834, A334741.
x- and y-coordinates are given by A174344 and A274923, respectively.

a(A033638(n)) = a(A002620(n)) for n > 1.

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)

A280027 Fill an infinite square array by following a spiral around the origin; in the central cell enter a(0)=1; thereafter, in the n-th cell, enter the sum of the entries of those earlier cells that can be seen from that cell.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 23, 42, 76, 146, 239, 441, 852, 1389, 2536, 4971, 9832, 15312, 27964, 54801, 108787, 169086, 308758, 603612, 1201837, 2397202, 3656904, 6687912, 13067709, 25998877, 51918269, 79176868, 144799285, 282915788, 562653823, 1124083053, 2246758839

Offset: 0

N. J. A. Sloane, Dec 24 2016

The spiral track being used here is the same as in A274640, except that the starting cell here is indexed 0 (as in A274641).
The central cell gets index 0 (and we fill it in with the value a(0)=1).
"Can be seen from" means "that are on the same row, column, diagonal, or antidiagonal as".

			The central portion of the spiral is:
.
     7----4----2
     |         |
    13    1----1  239
     |             |
    23---42---76--146
.
After the terms a(0) to a(8) of the spiral have been filled in, the next cell contains 76+42+23+1+4 = 146 = a(9).

Lars Blomberg, Table of n, a(n) for n = 0..3384

Cf. A274640, A274641, A278180.

More terms from Lars Blomberg, Dec 25 2016

A278619 Hexagonal spiral constructed on the nodes of the triangular net 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, 4, 5, 6, 7, 8, 10, 12, 15, 18, 22, 26, 31, 36, 42, 49, 56, 64, 72, 82, 94, 106, 121, 139, 157, 179, 205, 231, 262, 298, 334, 376, 425, 481, 537, 601, 673, 745, 827, 921, 1027, 1133, 1254, 1393, 1550, 1707, 1886, 2091, 2322, 2553, 2815, 3113, 3447, 3781, 4157, 4582, 5063, 5600

Offset: 0

Omar E. Pol, Nov 24 2016

nonn

To evaluate a(n) consider only the two largest neighbors of a(n) that are present in the spiral when a(n) should be a new term in the spiral.
For the same idea but for an right triangle see A278645; for a square spiral see A278180.
It appears that the same idea for an isosceles triangle and also for a square array gives A030237.

			Illustration of initial terms as a spiral:
.
.             18 - 15 - 12
.             /          \
.           22    3 - 2   10
.           /    /     \   \
.         26    4   1 - 1   8
.           \    \         /
.           31    5 - 6 - 7
.             \
.              36 - 42 - 49
.
a(16) = 36 because the sum of its two largest neighbors is 31 + 5 = 36.
a(17) = 42 because the sum of its two largest neighbors is 36 + 6 = 42.
a(18) = 49 because the sum of its two largest neighbors is 42 + 7 = 49.
a(19) = 56 because the sum of its two largest neighbors is 49 + 7 = 56.

Cf. A030237, A274920, A274921, A278180, A278181, A278645.

A334741 Fill an infinite square array by following a spiral around the origin; in the central cell enter a(0)=1; thereafter, in the n-th cell, enter the sum of the entries of those earlier cells that are in the same row or column as that cell.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 11, 21, 40, 47, 93, 180, 203, 397, 796, 1576, 1675, 3305, 6636, 13192, 14004, 27607, 55029, 110192, 220024, 226740, 450123, 898661, 1798700, 3594248, 3704800, 7354303, 14681369, 29349536, 58710640, 117394896, 119196748, 237492079

Offset: 0

Alec Jones and Peter Kagey, May 09 2020

nonn

The spiral track being used here is the same as in A274640, except that the starting cell here is indexed 0 (as in A274641).

The central cell gets index 0 (and we fill it in with the value a(0)=1).

			Spiral begins:
     3----2----1
     |         |
     5    1----1   47
     |              |
     8---11---21---40
a(11) = 47 = 1 + 1 + 5 + 40, the sum of the cells in its row and column.

Peter Kagey, Table of n, a(n) for n = 0..2499

Cf. A280027.
Cf. A180714, A214526, A274640, A278180, A304586, A307834, A334742.
x- and y-coordinates are given by A174344 and A274923, respectively.

\\ here P(n) returns A174344 and A274923 as pair.
P(n)={my(m=sqrtint(n), k=ceil(m/2)); n -= 4*k^2; if(n<0, if(n<-m, [k, 3*k+n], [-k-n, k]), if(nAndrew Howroyd, May 09 2020

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.

A334745 Starting with a(1) = a(2) = 1, proceed in a square spiral, computing each term as the sum of diagonally adjacent prior terms.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 3, 2, 3, 1, 1, 3, 2, 3, 1, 1, 3, 3, 3, 3, 1, 1, 3, 3, 3, 3, 1, 1, 4, 3, 6, 3, 4, 1, 1, 4, 3, 6, 3, 4, 1, 1, 4, 4, 6, 6, 4, 4, 1, 1, 4, 4, 6, 6, 4, 4, 1, 1, 5, 4, 10, 6, 10, 4, 5, 1, 1, 5, 4, 10, 6

Offset: 1

Alec Jones and Peter Kagey, May 09 2020

nonn

			Spiral begins:
... 3---3---3---3---1
                    |
1---1---2---2---1   1
|               |   |
2   1---1---1   1   3
|   |       |   |   |
2   1   1---1   2   2
|   |           |   |
1   1---2---1---1   3
|                   |
1---3---2---3---1---1
The last illustrated term above is a(35) = 3 = 2 + 1 because diagonally down-right is 2 and diagonally down-left is 1.

Peter Kagey, Table of n, a(n) for n = 1..10000
Peter Kagey, Bitmap illustrating the parity of the first 2^22 terms. (Even and odd numbers are represented with black and white pixels respectively.)

Cf. A141481, A278180, A334741, A334742.

The x- and y-coordinates at n-th step are A174344 and A274923 respectively.

Conjecture: a(2n-1) = A247976(n).

cp's OEIS Frontend

A334742 Pascal's spiral: starting with a(1) = 1, proceed in a square spiral, computing each term as the sum of horizontally and vertically adjacent prior terms.

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

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Maple

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A280027 Fill an infinite square array by following a spiral around the origin; in the central cell enter a(0)=1; thereafter, in the n-th cell, enter the sum of the entries of those earlier cells that can be seen from that cell.

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A278619 Hexagonal spiral constructed on the nodes of the triangular net in which each new term is the sum of its two largest neighbors in the structure.

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A334741 Fill an infinite square array by following a spiral around the origin; in the central cell enter a(0)=1; thereafter, in the n-th cell, enter the sum of the entries of those earlier cells that are in the same row or column as that cell.

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PARI

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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A334745 Starting with a(1) = a(2) = 1, proceed in a square spiral, computing each term as the sum of diagonally adjacent prior terms.

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