A033951 -id:A033951 - cp's OEIS frontend

A080335 Diagonal in square spiral or maze arrangement of natural numbers.

1, 5, 9, 17, 25, 37, 49, 65, 81, 101, 121, 145, 169, 197, 225, 257, 289, 325, 361, 401, 441, 485, 529, 577, 625, 677, 729, 785, 841, 901, 961, 1025, 1089, 1157, 1225, 1297, 1369, 1445, 1521, 1601, 1681, 1765, 1849, 1937, 2025, 2117, 2209, 2305, 2401, 2501

Offset: 0

Paul Barry, Mar 19 2003

Interleaves the odd squares A016754 with (1+4n^2), A053755.
Squares of positive integers (plus 1 if n is odd). - Wesley Ivan Hurt, Oct 10 2013
a(n) is the maximum total number of queens that can coexist without attacking each other on an [n+3] X [n+3] chessboard, when the lone queen is in the most vulnerable position on the board. Specifically, the lone queen will placed in any center position, facing an opponent's "army" of size a(n)-1 == A137932(n+2). - Bob Selcoe, Feb 12 2015
a(n) is also the edge chromatic number of the complement of the (n+2) X (n+2) rook graph. - Eric W. Weisstein, Jan 31 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Edge Chromatic Number
Eric Weisstein's World of Mathematics, Rook Complement Graph
Eric Weisstein's World of Mathematics, Rook Graph
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A081347, A081348.
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.

[(3+4*n+2*n^2-(-1)^n)/2: n in [0..50]]; // Vincenzo Librandi, Sep 06 2011

A080335:=n->(n mod 2) + (n+1)^2; seq(A080335(k),k=0..49); # Wesley Ivan Hurt, Oct 10 2013

With[{nn = 60}, Riffle[Range[1, nn, 2]^2, 4 Range[nn]^2 + 1]] (* Harvey P. Dale, Jan 29 2012 *)
LinearRecurrence[{2, 0, -2, 1}, {1, 5, 9, 17}, 60] (* Harvey P. Dale, Jan 29 2012 *)
Table[(3 + 4 n + 2 n^2 - (-1)^n)/2, {n, 0, 50}] (* Wesley Ivan Hurt, Oct 10 2013 *)
Table[Mod[n, 2] + (n + 1)^2, {n, 0, 20}] (* Eric W. Weisstein, Jan 31 2024 *)

a(n) = (3 + 4*n + 2*n^2 - (-1)^n)/2.
a(2*n) = A016754(n), a(2*n+1) = A053755(n+1).
E.g.f.: exp(x)*(2 + 3*x + x^2) - cosh(x). The sequence 1,1,5,9,... is given by n^2+(1+(-1)^n)/2 with e.g.f. exp(1+x+x^2)*exp(x)-sinh(x). - Paul Barry, Sep 02 2003 and Sep 19 2003
a(0)=1, a(1)=5, a(2)=9, a(3)=17, a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - Harvey P. Dale, Jan 29 2012
a(n)+(-1)^n = A137928(n+1). - Philippe Deléham, Feb 17 2012
G.f.: (1 + 3*x - x^2 + x^3)/((1-x)^3*(1+x)). - Colin Barker, Mar 18 2012
a(n) = A000035(n) + A000290(n+1). - Wesley Ivan Hurt, Oct 10 2013
From Bob Selcoe, Feb 12 2015: (Start)
a(n) = A137932(n+2) + 1.
a(n) = (n+1)^2 when n is even; a(n) = (n+1)^2 + 1 when n is odd.
a(n) = A002378(n+2) - A047238(n+3) + 1.
(End)
Sum_{n>=0} 1/a(n) = Pi*coth(Pi/2)/4 + Pi^2/8 - 1/2. - Amiram Eldar, Jul 07 2022

A137932 Terms in an n X n spiral that do not lie on its principal diagonals.

Original entry on oeis.org

0, 0, 0, 4, 8, 16, 24, 36, 48, 64, 80, 100, 120, 144, 168, 196, 224, 256, 288, 324, 360, 400, 440, 484, 528, 576, 624, 676, 728, 784, 840, 900, 960, 1024, 1088, 1156, 1224, 1296, 1368, 1444, 1520, 1600, 1680, 1764, 1848, 1936, 2024, 2116, 2208, 2304, 2400, 2500, 2600, 2704, 2808

Offset: 0

William A. Tedeschi, Feb 29 2008

The count of terms not on the principal diagonals is always even.
The last digit is the repeating pattern 0,0,0,4,8,6,4,6,8,4, which is palindromic if the leading 0's are removed, 4864684.
The sum of the last digits is 40, which is the count of the pattern times 4.
A 4 X 4 spiral is the only spiral, aside from a 0 X 0, whose count of terms that do not lie on its principal diagonals equal the count of terms that do [A137932(4) = A042948(4)] making the 4 X 4 the "perfect spiral".
Yet another property is mod(a(n), A042948(n)) = 0 iff n is even. This is a large family that includes the 4 X 4 spiral.
a(n) is the maximum number of queens of one color that can coexist without attacking one queen of the opponent's color on an [n+1] X [n+1] chessboard, when the lone queen is in the most vulnerable position on the board, i.e., on a center square. - Bob Selcoe, Feb 12 2015
Also the circumference of the (n-1) X (n-1) grid graph for n > 2. - Eric W. Weisstein, Mar 25 2018
Also the crossing number of the complete bipartite graph K_{5,n}. - Eric W. Weisstein, Sep 11 2018

			a(0) = 0^2 - (2(0) - mod(0,2)) = 0.
a(3) = 3^2 - (2(3) - mod(3,2)) = 4.

Enrique Pérez Herrero, Table of n, a(n) for n = 0..5000
Kival Ngaokrajang, Illustration of initial terms.
Eric Weisstein's World of Mathematics, Complete Bipartite Graph.
Eric Weisstein's World of Mathematics, Graph Circumference.
Eric Weisstein's World of Mathematics, Graph Crossing Number.
Eric Weisstein's World of Mathematics, Grid Graph.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A042948.
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.

A137932:=n->2*floor((n-1)^2/2); seq(A137932(n), n=0..50); # Wesley Ivan Hurt, Apr 19 2014

Table[2 Floor[(n - 1)^2/2], {n, 0, 20}] (* Enrique Pérez Herrero, Jul 04 2012 *)
2 Floor[(Range[20] - 1)^2/2] (* Eric W. Weisstein, Sep 11 2018 *)
Table[n^2 - 2 n + (1 - (-1)^n)/2, {n, 20}] (* Eric W. Weisstein, Sep 11 2018 *)
LinearRecurrence[{2, 0, -2, 1}, {0, 0, 4, 8}, 20] (* Eric W. Weisstein, Sep 11 2018 *)
CoefficientList[Series[-((4 x^2)/((-1 + x)^3 (1 + x))), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 11 2018 *)

A137932(n)={ return(n^2 - (2*n-n%2))} ;
print(vector(30,n,A137932(n-1))); /* R. J. Mathar, May 12 2014 */

Python
```
a = lambda n: n**2 - (2*n - (n%2))
```

a(n) = n^2 - (2*n - mod(n,2)) = n^2 - A042948(n).
a(n) = 2*A007590(n-1). - Enrique Pérez Herrero, Jul 04 2012
G.f.: -4*x^3 / ( (1+x)*(x-1)^3 ). a(n) = 4*A002620(n-1). - R. J. Mathar, Jul 06 2012
From Bob Selcoe, Feb 12 2015: (Start)
a(n) = (n-1)^2 when n is odd; a(n) = (n-1)^2 - 1 when n is even.
a(n) = A002378(n) - A047238(n+1). (End)
From Amiram Eldar, Mar 20 2022: (Start)
Sum_{n>=3} 1/a(n) = Pi^2/24 + 1/4.
Sum_{n>=3} (-1)^(n+1)/a(n) = Pi^2/24 - 1/4. (End)
E.g.f.: x*(x - 1)*cosh(x) + (x^2 - x + 1)*sinh(x). - Stefano Spezia, Oct 17 2022

A156859 The main column of a version of the square spiral.

Original entry on oeis.org

0, 3, 7, 14, 22, 33, 45, 60, 76, 95, 115, 138, 162, 189, 217, 248, 280, 315, 351, 390, 430, 473, 517, 564, 612, 663, 715, 770, 826, 885, 945, 1008, 1072, 1139, 1207, 1278, 1350, 1425, 1501, 1580, 1660, 1743, 1827, 1914, 2002, 2093, 2185, 2280, 2376, 2475, 2575

Offset: 0

Emilio Apricena (emilioapricena(AT)yahoo.it), Feb 17 2009

This spiral is sometimes called an Ulam spiral, but square spiral is a better name. - N. J. A. Sloane, Jul 27 2018
It is easy to see that the only two primes in the sequence are 3, 7. Therefore the primes of the version of Ulam spiral are divided into four parts (see also A035608): northeast (NE), northwest (NW), southwest (SW), and southeast (SE).
Number of pairs (x,y) having x and y of opposite parity with x in {0,...,n} and y in {0,...,2n}. - Clark Kimberling, Jul 02 2012
Partial Sums of A014601(n). - Wesley Ivan Hurt, Oct 11 2013

E. Apricena, A version of Ulam Spiral divided into four parts.
Minh Nguyen, 2-adic Valuations of Square Spiral Sequences, Honors Thesis, Univ. of Southern Mississippi (2021).
Marco Ripà, The n x n x n Points Problem Optimal Solution, viXra.org.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000290, A000384, A004526, A014601 (first differences), A115258.
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.

A156859:=n->n^2+n+floor((n+1)/2); seq(A156859(k), k=0..100); # Wesley Ivan Hurt, Oct 11 2013

Table[n^2 + n + Floor[(n+1)/2], {n, 0, 100}] (* Wesley Ivan Hurt, Oct 11 2013 *)

a(n) = n^2 + n + floor((n+1)/2) = A002378(n) + A004526(n+1) = A002620(n+1) + 3*A002620(n).
From R. J. Mathar, Feb 20 2009: (Start)
G.f.: x*(3+x)/((1+x)*(1-x)^3).
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). (End)
a(n-1) = floor(n/(e^(1/n)-1)). - Richard R. Forberg, Jun 19 2013
a(n) = A000290(n+1) + A004526(-n-1). - Wesley Ivan Hurt, Jul 15 2013
a(n) + a(n+1) = A014105(n+1). - R. J. Mathar, Jul 15 2013
a(n) = floor(A000384(n+1)/2). - Bruno Berselli, Nov 11 2013
E.g.f.: (x*(5 + 2*x)*cosh(x) + (1 + 5*x + 2*x^2)*sinh(x))/2. - Stefano Spezia, Apr 24 2024
Sum_{n>=1} 1/a(n) = 4/9 + 2*log(2) - Pi/3. - Amiram Eldar, Apr 26 2024

More terms added by Wesley Ivan Hurt, Oct 11 2013

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

A267682 a(n) = 2a(n-1) - 2a(n-3) + a(n-4) for n > 3, with initial terms 1, 1, 4, 8.

Original entry on oeis.org

1, 1, 4, 8, 15, 23, 34, 46, 61, 77, 96, 116, 139, 163, 190, 218, 249, 281, 316, 352, 391, 431, 474, 518, 565, 613, 664, 716, 771, 827, 886, 946, 1009, 1073, 1140, 1208, 1279, 1351, 1426, 1502, 1581, 1661, 1744, 1828, 1915, 2003, 2094, 2186, 2281, 2377, 2476

Offset: 0

Robert Price, Jan 19 2016

Also, total number of ON (black) cells after n iterations of the "Rule 201" elementary cellular automaton starting with a single ON (black) cell.

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Robert Price, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A267679.
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.

rule=201; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) nbc=Table[Total[catri[[k]]],{k,1,rows}]; (* Number of Black cells in stage n *) Table[Total[Take[nbc,k]],{k,1,rows}] (* Number of Black cells through stage n *)
LinearRecurrence[{2, 0, -2, 1}, {1, 1, 4, 8}, 60] (* Vincenzo Librandi, Jan 19 2016 *)

Vec((1-x+2*x^2+2*x^3)/((1-x)^3*(1+x)) + O(x^100)) \\ Colin Barker, Jan 19 2016

print([n*(n-1)+n//2+1 for n in range(51)]) # Karl V. Keller, Jr., Jul 14 2021

G.f.: (1 - x + 2*x^2 + 2*x^3) / ((1-x)^3*(1+x)). - Colin Barker, Jan 19 2016
a(n) = n*(n-1) + floor(n/2) + 1. - Karl V. Keller, Jr., Jul 14 2021
E.g.f.: (exp(x)*(2 + x + 2*x^2) - sinh(x))/2. - Stefano Spezia, Jul 16 2021

Edited by N. J. A. Sloane, Jul 25 2018, replacing definition with simpler formula provided by Colin Barker, Jan 19 2016.

A033952 Write 1,2,... in a clockwise spiral on a square lattice, writing each digit at a separate lattice point, starting with 1 at the origin and 2 at x=0, y=-1; sequence gives the numbers on the positive x-axis.

Original entry on oeis.org

1, 8, 6, 2, 3, 6, 6, 0, 3, 1, 8, 0, 2, 7, 1, 3, 9, 3, 4, 3, 9, 6, 0, 6, 8, 9, 6, 1, 2, 4, 2, 1, 5, 9, 4, 1, 0, 7, 7, 1, 7, 8, 0, 2, 6, 3, 4, 2, 7, 1, 8, 2, 0, 2, 2, 3, 5, 6, 6, 3, 2, 4, 1, 4, 1, 5, 6, 4, 2, 9, 2, 5, 5, 6, 8, 5, 0, 6, 4, 6, 7, 9, 0, 7, 6, 6, 7, 7, 7, 6, 4, 8, 0, 9, 1, 9, 5, 5, 9, 1, 0, 4, 0, 0, 9

Offset: 1

Olivier Gorin (gorin(AT)roazhon.inra.fr)

Same as the South spoke of the Champernowne spiral (A244677).

			The spiral begins
.
  3---1---4---1---5
  |               |
  1   5---6---7   1
  |   |       |   |
  2   4   1   8   6
  |   |   |   |   |
  1   3---2   9   1
  |           |   |
  1---1---0---1   7
.

Andrey Zabolotskiy, Table of n, a(n) for n = 1..10000

Sequences from spirals: A001107, A002939, A007742, A033951, A033952, A033953, A033954, A033989, A033990, A033991, A002943, A033996, A033988.

almostNatural[n_, b_] := Block[{m = 0, d = n, i = 1, l, p}, While[m <= d, l = m; m = (b - 1) i*b^(i - 1) + l; i++]; i--; p = Mod[d - l, i]; q = Floor[(d - l)/i] + b^(i - 1); If[p != 0, IntegerDigits[q, b][[p]], Mod[q - 1, b]]];
f[n_] := 4n^2 - 5n + 2; Array[ almostNatural[ f@#, 10] &, 105] (* Robert G. Wilson v, Aug 08 2014 *)

More terms from Andrew J. Gacek (andrew(AT)dgi.net)

Edited by Charles R Greathouse IV, Nov 01 2009

A033988 Write 0,1,2,... in a clockwise spiral on a square lattice, writing each digit at a separate lattice point, starting with 0 at the origin and 1 at x=0, y=-1; sequence gives the numbers on the positive y-axis.

Original entry on oeis.org

0, 5, 1, 4, 3, 7, 8, 0, 4, 7, 7, 1, 2, 6, 2, 1, 8, 7, 4, 2, 6, 1, 8, 9, 2, 7, 6, 0, 6, 5, 1, 2, 0, 4, 1, 5, 8, 5, 1, 8, 8, 8, 2, 1, 2, 3, 2, 4, 9, 0, 2, 8, 9, 9, 3, 3, 2, 0, 3, 7, 9, 3, 4, 2, 8, 8, 4, 7, 1, 5, 5, 3, 7, 4, 5, 9, 7, 5, 6, 5, 9, 8, 7, 1, 5, 3, 7, 8, 4, 0, 8, 5, 6, 9, 9, 3, 1, 0, 9, 8, 1, 1, 6, 9, 9

Offset: 0

N. J. A. Sloane

In other words, write 0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 ... in a clockwise spiral, starting with the 0 and taking the first step south; the sequence is then picked out from the resulting spiral by starting at the origin and moving north.

			  1---3---1---4---1
  |               |
  2   4---5---6   5
  |   |       |   |
  1   3   0   7   1
  |   |   |   |   |
  1   2---1   8   6
  |           |   |
  1---0---1---9   1
.
We begin with the 0 and wrap the numbers 1 2 3 4 ... around it.
Then the sequence is obtained by reading vertically upwards, starting from the initial 0.

Andrew Woods, Table of n, a(n) for n = 0..1000

Sequences based on the same spiral: A033953, A033989, A033990. Spiral without zero: A033952.
Other sequences from spirals: A001107, A002939, A007742, A033951, A033954, A033991, A002943, A033996.
Cf. A033307.

nmax = 105; A033307 = Flatten[IntegerDigits /@ Range[0, nmax^2 + 10 nmax]]; a[n_] := If[n==0, 0, A033307[[4n^2 + n + 1]]]; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Apr 24 2017, after Andrew Woods *)

a(n) = A033307(4*n^2 + n - 1) for n > 0. - Andrew Woods, May 18 2012

More terms from Andrew Gacek (andrew(AT)dgi.net)

Edited by Jon E. Schoenfield, Aug 12 2018

A033989 Write 0,1,2,... in a clockwise spiral on a square lattice, writing each digit at a separate lattice point, starting with 0 at the origin and 1 at x=0, y=-1; sequence gives the numbers on the negative x-axis.

Original entry on oeis.org

0, 3, 1, 1, 3, 2, 7, 9, 1, 1, 6, 9, 4, 7, 9, 1, 2, 1, 2, 1, 6, 7, 4, 3, 6, 1, 2, 9, 5, 1, 1, 0, 9, 3, 1, 3, 6, 6, 1, 8, 6, 9, 2, 5, 0, 2, 2, 4, 6, 6, 2, 5, 6, 0, 3, 8, 9, 5, 3, 3, 6, 9, 4, 0, 5, 4, 4, 9, 8, 0, 5, 0, 4, 5, 5, 3, 3, 1, 6, 8, 5, 8, 6, 5, 1, 4, 7, 4, 9, 1, 8, 5, 1, 9, 9, 8, 6, 6, 9, 1, 1, 6, 4, 8, 1

Offset: 0

N. J. A. Sloane

			  2---3---2---4---2---5---2
  |                       |
  2   1---3---1---4---1   6
  |   |               |   |
  2   2   4---5---6   5   2
  |   |   |       |   |   |
  1   1   3   0   7   1   7
  |   |   |   |   |   |   |
  2   1   2---1   8   6   2
  |   |           |   |   |
  0   1---0---1---9   1   8
  |                   |   |
  2---9---1---8---1---7   2
We begin with the 0 and wrap the numbers 1 2 3 4 ... around it. Then the sequence is obtained by reading leftwards, starting from the initial 0. - _Andrew Woods_, May 20 2012

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Sequences based on the same spiral: A033953, A033988, A033990. Spiral without zero: A033952.

Other sequences from spirals: A001107, A002939, A007742, A033951, A033954, A033991, A002943, A033996, A033988.

a033989 0 = 0
a033989 n = a033307 $ a185950 n  -- Reinhard Zumkeller, Aug 14 2013

nmax = 1000; A033307 = Flatten[IntegerDigits /@ Range[0, nmax^2+10*nmax]]; a[n_] := If[n==0, 0, A033307[[4n^2-n+1]]]; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Apr 23 2017, after Andrew Woods *)

a(n) = A033307(4*n^2-n-1) for n > 0. - Andrew Woods, May 20 2012

More terms from Andrew J. Gacek (andrew(AT)dgi.net)

Edited by Charles R Greathouse IV, Nov 01 2009

A115258 Isolated primes in Ulam's lattice (1, 2, ... in spiral).

Original entry on oeis.org

83, 101, 127, 137, 163, 199, 233, 311, 373, 443, 463, 491, 541, 587, 613, 631, 641, 659, 673, 683, 691, 733, 757, 797, 859, 881, 911, 919, 953, 971, 991, 1013, 1051, 1061, 1103, 1109, 1117, 1193, 1201, 1213, 1249, 1307, 1319, 1409, 1433, 1459, 1483, 1487

Offset: 1

Giorgio Balzarotti and Paolo P. Lava, Feb 17 2006

nonn

Isolated prime numbers have no adjacent primes in a lattice generated by writing consecutive integers starting from 1 in a spiral distribution. If n0 is the number of isolated primes and p the number of primes less than N, the ratio n0/p approaches 1 as N increases. If n1, n2, n3, n4 denote the number of primes with respectively 1, 2, 3, 4 adjacent primes in the lattice, the ratios n1/n0, n2/n1, n3/n2, n4/n3 approach 0 as N increases. The limits stand for any 2D lattice of integers generated by a priori criteria (i.e., not knowing distributions of primes) as Ulam's lattice.

			83 is an isolated prime as the adjacent numbers in lattice 50, 51, 81, 82, 84, 123, 124, 125 are not primes.
From _Michael De Vlieger_, Dec 22 2015: (Start)
Spiral including n <= 17^2 showing only primes, with the isolated primes in parentheses (redrawn by _Jon E. Schoenfield_, Aug 06 2017):
  257 .  .  .  .  . 251 .  .  .  .  .  .  .  .  . 241
   . 197 .  .  . 193 . 191 .  .  .  .  .  .  .  .  .
   .  .  .  .  .  .  .  . 139 .(137).  .  .  .  . 239
   .(199).(101).  .  . 97  .  .  .  .  .  .  . 181 .
   .  .  .  .  .  .  .  . 61  . 59  .  .  . 131 .  .
   .  .  . 103 . 37  .  .  .  .  . 31  . 89  . 179 .
  263 . 149 . 67  . 17  .  .  . 13  .  .  .  .  .  .
   .  .  .  .  .  .  .  5  .  3  . 29  .  .  .  .  .
   .  . 151 .  .  . 19  .  .  2 11  . 53  .(127).(233)
   .  .  . 107 . 41  .  7  .  .  .  .  .  .  .  .  .
   .  .  .  . 71  .  .  . 23  .  .  .  .  .  .  .  .
   .  .  . 109 . 43  .  .  . 47  .  .  .(83) . 173 .
  269 .  .  . 73  .  .  .  .  . 79  .  .  .  .  . 229
   .  .  .  .  . 113 .  .  .  .  .  .  .  .  .  .  .
  271 . 157 .  .  .  .  .(163).  .  . 167 .  .  . 227
   . 211 .  .  .  .  .  .  .  .  .  .  . 223 .  .  .
   .  .  .  . 277 .  .  . 281 . 283 .  .  .  .  .  .
(End)

G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 22.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Spiral.

Cf. A001107, A002939, A007742, A033951-A033954, A033989, A033990, A033991, A002943, A033996, A033988, A014848.

Cf. A113688 (isolated semiprimes in the semiprime spiral), A156859.

# A is Ulam's lattice
if (isprime(A[x,y])and(not(isprime(A[x+1,y]) or isprime(A[x-1,y])or isprime(A[x,y+1])or isprime(A[x,y-1])or isprime(A[x-1,y-1])or isprime(A[x+1,y+1])or isprime(A[x+1,y-1])or isprime(A[x-1,y+1])))) then print (A[x,y]) ; fi;

spiral[n_] := Block[{o = 2 n - 1, t, w}, t = Table[0, {o}, {o}]; t = ReplacePart[t, {n, n} -> 1]; Do[w = Partition[Range[(2 (# - 1) - 1)^2 + 1, (2 # - 1)^2], 2 (# - 1)] &@ k; Do[t = ReplacePart[t, {(n + k) - (j + 1), n + (k - 1)} -> #[[1, j]]]; t = ReplacePart[t, {n - (k - 1), (n + k) - (j + 1)} -> #[[2, j]]]; t = ReplacePart[t, {(n - k) + (j + 1), n - (k - 1)} -> #[[3, j]]]; t = ReplacePart[t, {n + (k - 1), (n - k) + (j + 1)} -> #[[4, j]]], {j, 2 (k - 1)}] &@ w, {k, 2, n}]; t]; f[w_] := Block[{d = Dimensions@ w, t, g}, t = Reap[Do[Sow@ Take[#[[k]], {2, First@ d - 1}], {k, 2, Last@ d - 1}]][[-1, 1]] &@ w; g[n_] := If[n != 0, Total@ Join[Take[w[[Last@ # - 1]], {First@ # - 1, First@ # + 1}], {First@ #, Last@ #} &@ Take[w[[Last@ #]], {First@ # - 1, First@ # + 1}], Take[w[[Last@ # + 1]], {First@ # - 1, First@# + 1}]] &@(Reverse@ First@ Position[t, n] + {1, 1}) == 0, False]; Select[Union@ Flatten@ t, g@ # &]]; f[spiral@ 21 /. n_ /; CompositeQ@ n -> 0] (* Michael De Vlieger, Dec 22 2015, Version 10 *)

A078510 Spiro-Fibonacci numbers, a(n) = sum of two previous terms that are nearest when terms arranged in a spiral.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 21, 24, 27, 31, 36, 42, 48, 54, 61, 69, 78, 88, 98, 108, 119, 131, 144, 158, 172, 186, 201, 217, 235, 256, 280, 304, 328, 355, 386, 422, 464, 512, 560, 608, 662, 723, 792, 870, 958, 1056

Offset: 0

Neil Fernandez, Jan 05 2003

nonn

Or "Spironacci numbers" for short. See also Spironacci polynomials, A265408. This sequence has an interesting growth rate, see A265370 and A265404. - Antti Karttunen, Dec 13 2015

			Terms are written in square boxes radiating spirally (cf. Ulam prime spiral). a(0)=0 and a(1)=1, so write 0 and then 1 to its right. a(2) goes in the box below a(1). The nearest two filled boxes contain a(0) and a(1), so a(2)=a(0)+a(1)=0+1=1. a(3) goes in the box to the left of a(2). The nearest two filled boxes contain a(0) and a(2), so a(3)=a(0)+a(2)=0+1=1.
From _Antti Karttunen_, Dec 17 2015: (Start)
The above description places cells in clockwise direction. However, for the computation of this sequence the actual orientation of the spiral is irrelevant. Following the convention used at A265409, we draw this spiral counterclockwise:
+--------+--------+--------+--------+
|a(15)   |a(14)   |a(13)   |a(12)   |
| = a(14)| = a(13)| = a(12)| = a(11)|
| + a(4) | + a(3) | + a(2) | + a(2) |
| = 9    | = 8    | = 7    | = 6    |
+--------+--------+--------+--------+
|a(4)    |a(3)    |a(2)    |a(11)   |
| = a(3) | = a(2) | = a(1) | = a(10)|
| + a(0) | + a(0) | + a(0) | + a(2) |
| = 1    | = 1    | = 1    | = 5    |
+--------+--------+--------+--------+
|a(5)    | START  |   ^    |a(10)   |
| = a(4) | a(0)=0 | a(1)=1 | = a(9) |
| + a(0) |   -->  |        | + a(1) |
| = 1    |        |        | = 4    |
+--------+--------+--------+--------+
|a(6)    |a(7)    |a(8)    |a(9)    |
| = a(5) | = a(6) | = a(7) | = a(8) |
| + a(0) | + a(0) | + a(1) | + a(1) |
| = 1    | = 1    | = 2    | = 3    |
+--------+--------+--------+--------+
(End)

Antti Karttunen, Table of n, a(n) for n = 0..1024

Cf. A000045, A001222, A033951, A063826, A265407, A265408, A265409, A265359, A265370, A265404.

From Antti Karttunen, Dec 13 2015: (Start)
a(0) = 0, a(1) = 1; for n > 1, a(n) = a(n-1) + a(A265409(n)).
equally, for n > 1, a(n) = a(n-1) + a(n - A265359(n)).
a(n) = A001222(A265408(n)).
(End)

cp's OEIS Frontend

A080335 Diagonal in square spiral or maze arrangement of natural numbers.

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A137932 Terms in an n X n spiral that do not lie on its principal diagonals.

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A156859 The main column of a version of the square spiral.

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

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A267682 a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n > 3, with initial terms 1, 1, 4, 8.

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A033952 Write 1,2,... in a clockwise spiral on a square lattice, writing each digit at a separate lattice point, starting with 1 at the origin and 2 at x=0, y=-1; sequence gives the numbers on the positive x-axis.

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A033988 Write 0,1,2,... in a clockwise spiral on a square lattice, writing each digit at a separate lattice point, starting with 0 at the origin and 1 at x=0, y=-1; sequence gives the numbers on the positive y-axis.

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A267682 a(n) = 2a(n-1) - 2a(n-3) + a(n-4) for n > 3, with initial terms 1, 1, 4, 8.