A274640 -id:A274640 - cp's OEIS frontend

A274927 Northwest spoke of spiral in A274640.

1, 2, 3, 4, 8, 9, 7, 11, 14, 10, 22, 29, 18, 19, 25, 26, 20, 30, 33, 28, 31, 32, 45, 67, 41, 34, 38, 44, 48, 51, 49, 100, 55, 52, 58, 53, 60, 61, 64, 63, 121, 75, 70, 65, 71, 72, 83, 81, 74, 79, 84, 82, 86, 88, 85, 87, 176, 95, 96, 93, 106, 103, 109, 112, 105

Offset: 0

N. J. A. Sloane, Jul 12 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.

Cf. A274640, A274641. The 8 spokes are A274924-A274931.

More terms from Alois P. Heinz, Jul 12 2016

A274929 Southwest spoke of spiral in A274640.

Original entry on oeis.org

1, 4, 6, 5, 14, 10, 11, 23, 16, 18, 21, 24, 19, 29, 46, 44, 30, 31, 58, 36, 33, 43, 41, 39, 40, 79, 78, 83, 47, 91, 57, 62, 56, 54, 61, 63, 59, 66, 115, 65, 73, 72, 75, 76, 131, 144, 92, 84, 81, 71, 82, 87, 164, 100, 172, 106, 104, 174, 179, 182, 105, 101, 191

Offset: 0

N. J. A. Sloane, Jul 12 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.

Cf. A274640, A274641. The 8 spokes are A274924-A274931.

More terms from Alois P. Heinz, Jul 12 2016

A308897 Walk a rook along the square spiral numbered 1, 2, 3, ... (cf. A274640); a(n) = mex of earlier values the rook can move to.

Original entry on oeis.org

1, 2, 1, 2, 3, 4, 1, 3, 4, 2, 3, 4, 1, 3, 4, 2, 5, 6, 7, 8, 1, 5, 6, 7, 8, 2, 5, 6, 7, 8, 1, 5, 6, 7, 8, 2, 3, 6, 5, 8, 7, 4, 1, 3, 6, 5, 8, 7, 4, 2, 3, 6, 5, 8, 7, 4, 1, 3, 6, 5, 8, 7, 4, 2, 9, 10, 11, 12, 13, 14, 15, 16, 1, 9, 10, 11, 12, 13, 14, 15, 16, 2

Offset: 1

N. J. A. Sloane, Jul 02 2019

nonn

Mex here means "smallest positive missing number".

Add 1 to the terms of A308896.

Cf. A274640, A308896.

More terms from Rémy Sigrist, Jul 02 2019

A317186 One of many square spiral sequences: a(n) = n^2 + n - floor((n-1)/2).

Original entry on oeis.org

1, 2, 6, 11, 19, 28, 40, 53, 69, 86, 106, 127, 151, 176, 204, 233, 265, 298, 334, 371, 411, 452, 496, 541, 589, 638, 690, 743, 799, 856, 916, 977, 1041, 1106, 1174, 1243, 1315, 1388, 1464, 1541, 1621, 1702, 1786, 1871, 1959, 2048, 2140, 2233, 2329, 2426

Offset: 0

N. J. A. Sloane, Jul 27 2018

Draw a square spiral on a piece of graph paper, and label the cells starting at the center with the positive (resp. nonnegative) numbers. This produces two versions of the labeled square spiral, shown in the Example section below.
The spiral may proceed clockwise or counterclockwise, and the first arm of the spiral may be along any of the four axes, so there are eight versions of each spiral. However, this has no effect on the resulting sequences, and it is enough to consider just two versions of the square spiral (starting at 1 or starting at 0).
The present sequence is obtained by reading alternate entries on the X-axis (say) of the square spiral started at 1.
The cross-references section lists many sequences that can be read directly off the two spirals. Many other sequences can be obtained from them by using them to extract subsequences from other important sequences. For example, the subsequence of primes indexed by the present sequence gives A317187.
a(n) is also the number of free polyominoes with n + 4 cells whose difference between length and width is n. In this comment the length is the longer of the two dimensions and the width is the shorter of the two dimensions (see the examples of polyominoes). Hence this is also the diagonal 4 of A379625. - Omar E. Pol, Jan 24 2025
From John Mason, Feb 19 2025: (Start)
The sequence enumerates polyominoes of width 2 having precisely 2 horizontal bars. By classifying such polyominoes according to the following templates, it is possible to define a formula that reduces to the one below:
.
 OO  O  O
 O  OO OO
 O  O  O
 O  O  OO
 OO OO  O
.
(End)

			The square spiral when started with 1 begins:
.
  100--99--98--97--96--95--94--93--92--91
                                        |
   65--64--63--62--61--60--59--58--57  90
    |                               |   |
   66  37--36--35--34--33--32--31  56  89
    |   |                       |   |   |
   67  38  17--16--15--14--13  30  55  88
    |   |   |               |   |   |   |
   68  39  18   5---4---3  12  29  54  87
    |   |   |   |       |   |   |   |   |
   69  40  19   6   1---2  11  28  53  86
    |   |   |   |           |   |   |   |
   70  41  20   7---8---9--10  27  52  85
    |   |   |                   |   |   |
   71  42  21--22--23--24--25--26  51  84
    |   |                           |   |
   72  43--44--45--46--47--48--49--50  83
    |                                   |
   73--74--75--76--77--78--79--80--81--82
.
For the square spiral when started with 0, subtract 1 from each entry. In the following diagram this spiral has been reflected and rotated, but of course that makes no difference to the sequences:
.
   99  64--65--66--67--68--69--70--71--72
    |   |                               |
   98  63  36--37--38--39--40--41--42  73
    |   |   |                       |   |
   97  62  35  16--17--18--19--20  43  74
    |   |   |   |               |   |   |
   96  61  34  15   4---5---6  21  44  75
    |   |   |   |   |       |   |   |   |
   95  60  33  14   3   0   7  22  45  76
    |   |   |   |   |   |   |   |   |   |
   94  59  32  13   2---1   8  23  46  77
    |   |   |   |           |   |   |   |
   93  58  31  12--11--10---9  24  47  78
    |   |   |                   |   |   |
   92  57  30--29--28--27--26--25  48  79
    |   |                           |   |
   91  56--55--54--53--52--51--50--49  80
    |                                   |
   90--89--88--87--86--85--84--83--82--81
.
From _Omar E. Pol_, Jan 24 2025: (Start)
For n = 0 there is only one free polyomino with 0 + 4 = 4 cells whose difference between length and width is 0 as shown below, so a(0) = 1.
   _ _
  |_|_|
  |_|_|
.
For n = 1 there are two free polyominoes with 1 + 4 = 5 cells whose difference between length and width is 1 as shown below, so a(1) = 2.
   _ _     _ _
  |_|_|   |_|_|
  |_|_|   |_|_
  |_|     |_|_|
.
(End)

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
Index entries for sequences related to polyominoes.

Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
Filling in these two squares spirals with greedy algorithm: A274640, A274641.
Cf. also A317187.
Cf. A000105, A379625.

a[n_] := n^2 + n - Floor[(n - 1)/2]; Array[a, 50, 0] (* Robert G. Wilson v, Aug 01 2018 *)
LinearRecurrence[{2, 0, -2 , 1},{1, 2, 6, 11},50] (* or *)
CoefficientList[Series[(- x^3 - 2 * x^2 - 1) / ((x - 1)^3 * (x + 1)), {x, 0, 50}], x] (* Stefano Spezia, Sep 02 2018 *)

From Daniel Forgues, Aug 01 2018: (Start)
a(n) = (1/4) * (4 * n^2 + 2 * n + (-1)^n + 3), n >= 0.
a(0) = 1; a(n) = - a(n-1) + 2 * n^2 - n + 2, n >= 1.
a(0) = 1; a(1) = 2; a(2) = 6; a(3) = 11; a(n) = 2 * a(n-1) - 2 * a(n-3) + a(n-4), n >= 4.
G.f.: (- x^3 - 2 * x^2 - 1) / ((x - 1)^3 * (x + 1)). (End)
E.g.f.: ((2 + 3*x + 2*x^2)*cosh(x) + (1 + 3*x + 2*x^2)*sinh(x))/2. - Stefano Spezia, Apr 24 2024
a(n)+a(n+1)=A033816(n). - R. J. Mathar, Mar 21 2025
a(n)-a(n-1) = A042948(n), n>=1. - R. J. Mathar, Mar 21 2025

A274821 Hexagonal spiral constructed on the nodes of the infinite triangular net in which each term is the least positive integer such that no diagonal contains a repeated term.

Original entry on oeis.org

1, 2, 3, 2, 3, 2, 3, 1, 4, 1, 5, 4, 6, 4, 5, 6, 4, 5, 7, 6, 7, 8, 5, 7, 6, 8, 7, 4, 1, 7, 6, 8, 1, 7, 8, 6, 5, 9, 2, 4, 7, 9, 2, 10, 8, 9, 3, 5, 10, 9, 3, 11, 12, 9, 10, 11, 13, 4, 9, 10, 8, 11, 10, 3, 5, 9, 6, 11, 3, 12, 10, 12, 1, 11, 8, 9, 7, 1, 10, 8, 11, 13, 8, 2, 5, 9, 6, 12, 2, 11, 13, 10, 6, 12, 11, 14, 13, 12, 14, 15

Offset: 0

Omar E. Pol, Jul 09 2016

nonn

Also spiral constructed on the infinite hexagonal grid in which each term is the least positive integer such that no diagonal of successive adjacent cells contains a repeated term. Every number is located in the center of a hexagonal cell. Every cell is also the center of three diagonals of successive adjacent cells.

			Illustration of initial terms as a spiral:
.
.                  10 - 5 - 3 - 9 - 8
.                  /                 \
.                 9   4 - 7 - 8 - 6   10
.                /   /             \   \
.               3   1   6 - 4 - 5   7   2
.              /   /   /         \   \   \
.            11   7   4   2 - 3   1   5   9
.            /   /   /   /     \   \   \   \
.          12   6   5   3   1 - 2   4   8   7
.            \   \   \   \         /   /   /
.             9   8   6   2 - 3 - 1   7   4
.              \   \   \             /   /
.              10   1   4 - 5 - 7 - 6   2
.                \   \                 /
.                11   7 - 8 - 6 - 5 - 9
.                  \
.                  13 - 4 - 9 - 10 - 8
.

Cf. A269526 (square array), A274640 (square spiral), A274651 (right triangle), A274820, A274920, A274921, A275606, A275610.

a(n) = A274820(n) + 1.

A324774 East spoke of spiral in A274641.

Original entry on oeis.org

0, 1, 3, 7, 10, 11, 15, 8, 18, 23, 21, 17, 26, 25, 20, 36, 42, 38, 39, 48, 27, 28, 31, 45, 54, 59, 44, 47, 65, 72, 69, 75, 82, 76, 64, 74, 41, 61, 93, 95, 100, 102, 66, 62, 111, 79, 112, 57, 106, 63, 107, 119, 108, 68, 123, 129, 139

Offset: 0

N. J. A. Sloane, Jul 05 2019

nonn

F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.

Cf. A274640, A274641.

The 8 spokes are A324774-A324781. To get them, subtract 1 from A274924-A274931, respectively.

A324781 South-East spoke of spiral in A274641.

Original entry on oeis.org

0, 5, 4, 11, 15, 16, 20, 23, 26, 12, 14, 35, 39, 42, 46, 53, 22, 55, 58, 61, 38, 67, 68, 34, 45, 36, 41, 89, 49, 96, 93, 98, 101, 56, 106, 114, 116, 115, 119, 65, 129, 130, 72, 76, 139, 142, 75, 79, 77, 158, 160, 155, 164, 91, 90, 173, 88, 180, 103, 186, 189

Offset: 0

N. J. A. Sloane, Jul 05 2019

nonn

F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.

Cf. A274640, A274641.

The 8 spokes are A324774-A324781. To get them, subtract 1 from A274924-A274931, respectively.

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.

Original entry on oeis.org

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.

A355270 Lexicographically earliest sequence of positive integers on a square spiral such that the sum of adjacent pairs of numbers within each row, column and diagonal is distinct in that row, column and diagonal.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 2, 4, 3, 3, 4, 4, 3, 5, 4, 2, 4, 3, 5, 4, 4, 2, 3, 6, 4, 6, 5, 7, 6, 2, 6, 3, 2, 5, 8, 4, 3, 6, 6, 7, 3, 5, 7, 6, 8, 8, 7, 1, 2, 7, 5, 1, 2, 5, 8, 6, 4, 8, 5, 6, 9, 7, 1, 4, 10, 1, 1, 6, 3, 9, 12, 5, 1, 7, 2, 1, 6, 4, 1, 13, 6, 4, 7, 9, 12, 10, 7, 11, 1, 5, 2, 10, 7, 4, 5, 8

Offset: 1

Scott R. Shannon, Jun 26 2022

nonn

In the first 2 million terms the largest number is 1959, while the number 1, the most commonly occurring number, appears 10893 times. See the linked images.

			The spiral begins:
.
                                .
    4---8---5---2---3---6---2   :
    |                       |   :
    3   2---4---5---3---4   6   5
    |   |               |   |   |
    6   4   2---1---1   4   7   2
    |   |   |       |   |   |   |
    6   3   2   1---1   3   5   1
    |   |   |           |   |   |
    7   5   3---2---4---3   6   5
    |   |                   |   |
    3   4---4---2---3---6---4   7
    |                           |
    5---7---6---8---8---7---1---2
.
a(25) = 6 as when a(25) is placed, at coordinate (2,-2) relative to the starting square, its adjacent squares are a(10) = 3, a(9) = 4, a(24) = 3. The sums of adjacent pairs of numbers in a(25)'s column are 3 + 3 = 6, 3 + 4 = 7, 4 + 4 = 8, in its northwest diagonal are 4 + 1 = 5, 1 + 2 = 3, 2 + 2 = 4, and in its row are 3 + 2 = 5, 2 + 4 = 6, 4 + 4 = 8. Setting a(25) to 1 would create a sum of 5 with its diagonal neighbor 4, but 5 has already occurred as a sum on this diagonal. Similarly numbers 2, 3, 4 and 5 can be eliminated as they create sums with the three adjacent numbers, 3, 4, and 3, which have already occurred along the corresponding column, diagonal or row. This leaves 6 as the smallest number which creates new sums, namely 9, 10 and 9, with its three neighbors that have not already occurred along the corresponding column, diagonal and row.

Scott R. Shannon, Image of the first 2 million terms. The values are scaled across the spectrum from red to violet, with the value ranges increasing towards the violet end to give more color weighting to the larger numbers.
Scott R. Shannon, Distribution of a(n) for the first 2 million terms. The number 1 appears 10893 times. The second maximum occurs at n ~ 500.

Cf. A355271, A274640, A275609, A307834.

A275915 First differences of A273059.

Original entry on oeis.org

9, 4, 4, 4, 61, 10, 10, 10, 116, 16, 16, 16, 169, 22, 22, 22, 119, 26, 26, 26, 259, 32, 32, 32, 314, 38, 38, 38, 367, 44, 44, 44, 421, 50, 50, 50, 476, 56, 56, 56, 529, 62, 62, 62, 319, 66, 66, 66, 619, 72, 72, 72, 674, 78, 78, 78, 727, 84, 84, 84, 782, 90, 90, 90, 835, 96, 96, 96, 489, 100

Offset: 0

N. J. A. Sloane, Aug 28 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..30287 (first 1408 terms from N. J. A. Sloane)
Alois P. Heinz, Positions of first 1409 1's in plane of A274640
Zak Seidov, Graph of first differences of A273059.
Zak Seidov, Spiral A274640 with ones shown.

Cf. A273059, A274640.

cp's OEIS Frontend

A274927 Northwest spoke of spiral in A274640.

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A274929 Southwest spoke of spiral in A274640.

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A308897 Walk a rook along the square spiral numbered 1, 2, 3, ... (cf. A274640); a(n) = mex of earlier values the rook can move to.

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A317186 One of many square spiral sequences: a(n) = n^2 + n - floor((n-1)/2).

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A274821 Hexagonal spiral constructed on the nodes of the infinite triangular net in which each term is the least positive integer such that no diagonal contains a repeated term.

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A324774 East spoke of spiral in A274641.

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A324781 South-East spoke of spiral in A274641.

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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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A355270 Lexicographically earliest sequence of positive integers on a square spiral such that the sum of adjacent pairs of numbers within each row, column and diagonal is distinct in that row, column and diagonal.

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A275915 First differences of A273059.

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