A054554 -id:A054554 - cp's OEIS frontend

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

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.

A073337 Primes of the form 4k^2 - 10k + 7 with k positive.

Original entry on oeis.org

3, 13, 31, 241, 307, 463, 757, 1123, 1723, 3307, 3541, 4831, 5113, 5701, 6007, 8011, 9901, 10303, 11131, 12433, 13807, 14281, 17293, 20023, 20593, 21757, 23563, 24181, 26083, 28057, 30103, 35911, 41413, 43891, 46441, 53593, 60271, 78121, 82657, 86143, 95791, 108571, 123553, 127807, 136531, 145543, 147073, 156421

Offset: 1

Zak Seidov, Aug 25 2002

Primes of the form k^2 + k + 1 with k odd and positive. - Peter Munn, Jan 27 2018

Primes of the form A000217(2*k) + A000217(2*k+2). - J. M. Bergot, May 09 2018

			3 is a term because for k=2, 4*k^2 - 10*k + 7 = 3 a prime.
7 is not a term because 7 can only be obtained with k=0 or k=5/2.

Muniru A Asiru, Table of n, a(n) for n = 1..5000

Cf. A000217, A054554, A073338, A168026.

Subset of A002383.

Filtered(List([2..300],n->4*n^2-10*n+7),IsPrime); # Muniru A Asiru, Apr 15 2018

[a: n in [1..400] | IsPrime(a) where a is 4*n^2 - 10*n + 7]; // Vincenzo Librandi, Dec 23 2019

select(isprime, [seq(4*n^2-10*n+7 ,n=2..300)]); # Muniru A Asiru, Apr 15 2018

Select[Table[4 n^2 - 10 n + 7, {n, 1, 200}], PrimeQ] (* Vincenzo Librandi, Dec 23 2019 *)

select(isprime,vector(300,k,4*k^2 - 10*k + 7)) \\ Joerg Arndt, Feb 28 2018

a(n) = A054554(A073338(n)). - Elmo R. Oliveira, Apr 20 2025

Edited by Dean Hickerson, Aug 28 2002
a(1)=7 inserted and typo in Mathematica code corrected by Vincenzo Librandi, Dec 09 2011
Incorrect term 7 removed by Joerg Arndt, Feb 28 2018

A227776 a(n) = 6*n^2 + 1.

Original entry on oeis.org

1, 7, 25, 55, 97, 151, 217, 295, 385, 487, 601, 727, 865, 1015, 1177, 1351, 1537, 1735, 1945, 2167, 2401, 2647, 2905, 3175, 3457, 3751, 4057, 4375, 4705, 5047, 5401, 5767, 6145, 6535, 6937, 7351, 7777, 8215, 8665, 9127, 9601, 10087, 10585, 11095, 11617, 12151

Offset: 0

Clark Kimberling, Jul 30 2013

Least splitter is defined for x < y at A227631 as the least positive integer d such that x <= c/d < y for some integer c; the number c/d is called the least splitting rational of x and y. Conjecture: a(n) is the least splitter of s(n) and s(n+1), where s(n) = n*sin(1/n).

			The first eight least splitting rationals for {n*sin(1/n), n >=1 } are these fractions: 6/7, 24/25, 54/55, 96/97, 150/151, 216/217, 294/295, 384/385.

Clark Kimberling, Table of n, a(n) for n = 0..1000
Leo Tavares, Illustration: Hexagonal Star Rays.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A227631, A287326.

Cf. A003154, A008458, A003215, A000217, A028896, A054554, A046092, A080855, A045943, A172043, A002378, A033581.

z = 40; r[x_, y_] := Module[{c, d}, d = NestWhile[#1 + 1 &, 1, ! (c = Ceiling[#1 x - 1]) < Ceiling[#1 y] - 1 &]; (c + 1)/d]; s[n_] := s[n] = n*Sin[1/n]; t = Table[r[s[n], s[n + 1]], {n, 1, z}] (* least splitting rationals *); fd = Denominator[t] (* Peter J. C. Moses, Jul 15 2013 *)
Array[6 #^2 + 1 &, 45] (* Michael De Vlieger, Nov 08 2017 *)
LinearRecurrence[{3,-3,1},{7,25,55},50] (* Harvey P. Dale, Dec 16 2017 *)

a(n)=6*n^2+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: (1 + 4*x + 7*x^2)/(1 - x)^3.
a(n) = A287326(2n, n). - Kolosov Petro, Nov 06 2017
From Amiram Eldar, Jul 15 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + (Pi/sqrt(6))*coth(Pi/sqrt(6)))/2.
Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/sqrt(6))*csch(Pi/sqrt(6)))/2. (End)
From Amiram Eldar, Feb 05 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/sqrt(6))*sinh(Pi/sqrt(3)).
Product_{n>=1} (1 - 1/a(n)) = (Pi/sqrt(6))*csch(Pi/sqrt(6)).(End)
From Leo Tavares, Nov 20 2021: (Start)
a(n) = A003154(n+1) - A008458(n). See Hexagonal Star Rays illustration.
a(n) = A003215(n) + A028896(n-1).
a(n) = A054554(n+1) + A046092(n).
a(n) = A080855(n) + A045943(n).
a(n) = A172043(n) + A002378(n).
a(n) = A033581(n) + 1. (End)
E.g.f.: exp(x)*(1 + 6*x + 6*x^2). - Stefano Spezia, Sep 14 2024

a(0) = 1 prepended by Robert P. P. McKone, Oct 09 2023

A054553 Prime number spiral (clockwise, Northeast spoke).

Original entry on oeis.org

2, 5, 41, 127, 269, 467, 751, 1093, 1523, 2027, 2621, 3299, 4007, 4861, 5749, 6763, 7867, 9041, 10273, 11719, 13121, 14723, 16319, 18061, 19963, 21851, 23857, 26021, 28289, 30661, 33029, 35531, 38201, 40993, 43759, 46751, 49789, 52957, 56197

Offset: 0

Enoch Haga and G. L. Honaker, Jr., Apr 10 2000

8-spoke wheel overlays prime number spiral; hub is 2 in shell 0; 8 spokes radiate from this hub; this is Northeast, clockwise.

			Begin a prime number spiral at shell 0 (prime 2), proceed clockwise, Northeast.
From _Omar E. Pol_, Feb 19 2022: (Start)
The spiral with four terms in every spoke looks like this:
.
  227  101--103--107--109--113--127
   |     |                       |
  223   97   29---31---37---41  131
   |     |    |              |   |
  211   89   23    3----5   43  137
   |     |    |    |    |    |   |
  199   83   19    2    7   47  139
   |     |    |         |    |   |
  197   79   17---13---11   53  149
   |     |                   |   |
  193   73---71---67---61---59  151
   |                             |
  191--181--179--173--167--163--157
.
(End)

Ivan Panchenko, Table of n, a(n) for n = 0..1000

Cf. A000040, A054554, A054551, A054555, A054564, A054566, A054568, A054570, A053999.

[NthPrime(4*n^2 - 10*n + 7): n in [1..40]]; // Vincenzo Librandi, Aug 29 2018

Table[ Prime[4n^2 - 10n + 7], {n, 1, 40} ]

a(n) = A000040(A054554(n+1)). - R. J. Mathar, Aug 29 2018

Edited by Robert G. Wilson v, Feb 25 2002

A114254 Sum of all terms on the two principal diagonals of a 2n+1 X 2n+1 square spiral.

Original entry on oeis.org

1, 25, 101, 261, 537, 961, 1565, 2381, 3441, 4777, 6421, 8405, 10761, 13521, 16717, 20381, 24545, 29241, 34501, 40357, 46841, 53985, 61821, 70381, 79697, 89801, 100725, 112501, 125161, 138737, 153261, 168765, 185281, 202841, 221477, 241221

Offset: 0

William A. Tedeschi, Feb 06 2008, Mar 01 2008

			For n = 1, the 3 X 3 spiral is
.
       7---8---9
       |
       6   1---2
       |       |
       5---4---3
.
so a(1) = 7 + 9 + 1 + 5 + 3 = 25.
.
For n = 2, the 5 X 5 spiral is
.
  21--22--23--24--25
   |
  20   7---8---9--10
   |   |           |
  19   6   1---2  11
   |   |       |   |
  18   5---4---3  12
   |               |
  17--16--15--14--13
.
so a(2) = 21 + 25 + 7 + 9 + 1 + 5 + 3 + 17 + 13 = 101.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Project Euler, Problem 28. Number Spiral Diagonals
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A016754, A054569, A053755, A054554 for diagonals from origin.

Cf. A325958 (first differences).

Array[1 + 10 #^2 + (16 #^3 + 26 #)/3 &, 36, 0] (* Michael De Vlieger, Mar 01 2018 *)

a(n) = 1 + 10*n^2 + (16*n^3 + 26*n)/3; \\ Joerg Arndt, Mar 01 2018

O.g.f.: 3/(-1+x) + 16/(-1+x)^2 + 44/(-1+x)^3 + 32/(-1+x)^4 = (1 + 21*x + 7*x^2 + 3*x^3)/(-1+x)^4. - R. J. Mathar, Feb 10 2008

a(n) = 1 + 10*n^2 + (16*n^3 + 26*n)/3. [Corrected by Arie Groeneveld, Aug 17 2008]

A200975 Numbers on the diagonals in Ulam's spiral.

Original entry on oeis.org

1, 3, 5, 7, 9, 13, 17, 21, 25, 31, 37, 43, 49, 57, 65, 73, 81, 91, 101, 111, 121, 133, 145, 157, 169, 183, 197, 211, 225, 241, 257, 273, 289, 307, 325, 343, 361, 381, 401, 421, 441, 463, 485, 507, 529, 553, 577, 601, 625, 651, 677, 703, 729, 757, 785, 813, 841, 871, 901

Offset: 1

Ismael Bouya, Nov 25 2011

All entries are odd.
From Bob Selcoe, Oct 22 2014: (Start)
The following hold:
1. a(n) = (2k + 1)^2  when n = 4k + 1, k >= 0
2. a(n) = 4*k^2 + 1   when n = 4k - 1, k > 0
3  a(n) = k^2 + k + 1 when n = 2k, k > 0.
Conjecture 1: there must be at least one prime in [a(n), a(n+1)] inclusive.
Conjecture 2: generally, when j is in [(2m-1)^2+1, (2m+1)^2] inclusive, there must be at least one prime in [j-2m-1, j] inclusive. If true, then Conjecture 1 is true; also suggests A248623, A248835 and Oppermann's conjecture (see A002620) likely are true. (End)

			The numbers between ** are in this sequence.
.
  *21*--22---23---24--*25*
    |
    |
   20   *7*---8---*9*--10
    |    |              |
    |    |              |
   19    6   *1*---2   11
    |    |         |    |
    |    |         |    |
   18   *5*---4---*3*  12
    |                   |
    |                   |
  *17*--16---15---14--*13*

Todd Silvestri, Table of n, a(n) for n = 1..1000
Project Euler, Problem 28: Number spiral diagonals
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

Cf. A016754, A054554, A053755, and A054569 interleaved, A002620,

Cf. A121658 (complementary)

Sort@ Flatten@ Table[4n^2 + (2j - 4)n + 1, {j, 0, 3}, {n, 16}] (* Robert G. Wilson v, Jul 10 2014 *)
a[n_Integer/;n>0]:=Quotient[2 n (n+2)+(-1)^n-4 Mod[n^2 (3 n+2),4,-1]+7,8] (* Todd Silvestri, Oct 25 2014 *)

al(n)=local(r=vector(n),j);r[1]=1;for(k=2,n,r[k]=r[k-1]+(k+2)\4*2);r /* Franklin T. Adams-Watters, Nov 26 2011 */

# prints all numbers on the diagonals of a sq*sq spiral
sq = 5
d = 1
while 2*d - 1 < sq:
    print(4*d*d - 4*d +1)
    print(4*d*d - 4*d +1 + 1* 2* d)
    print(4*d*d - 4*d +1 + 2* 2* d)
    print(4*d*d - 4*d +1 + 3* 2* d)
    d += 1
print(sq*sq)

a(4n) = 4n^2 + 2n + 1; a(4n+1) = 4n^2 + 4n + 1; a(4n+2) = 4n^2 + 6n + 3; a(4n+3) = 4n^2 + 8n + 5. [corrected by James Mitchell, Dec 31 2017]
G.f.: -x*(1+x+x^5-x^4) / ( (1+x)*(x^2+1)*(x-1)^3 ). - R. J. Mathar, Nov 28 2011
a(n) = (2*n*(n+2)+(-1)^n-4*sin((Pi*n)/2)+7)/8 = (A249356(n)+7)/8. - Todd Silvestri, Oct 25 2014
a(n) = floor_(n*(n+2)/4) + floor_(n(mod 4)/3) + 1. - Bob Selcoe, Oct 27 2014

Edited with more terms by Franklin T. Adams-Watters, Nov 26 2011

A081347 First column in maze arrangement of natural numbers.

Original entry on oeis.org

1, 2, 3, 12, 13, 30, 31, 56, 57, 90, 91, 132, 133, 182, 183, 240, 241, 306, 307, 380, 381, 462, 463, 552, 553, 650, 651, 756, 757, 870, 871, 992, 993, 1122, 1123, 1260, 1261, 1406, 1407, 1560, 1561, 1722, 1723, 1892, 1893, 2070, 2071, 2256, 2257, 2450, 2451

Offset: 0

Paul Barry, Mar 19 2003

Interleaves two times the hexagonal numbers A000384 with A054554.

			Starting with 1,2,3, turn (LL) and then repeat (RRR)(LLL) to get
1 6 7 20
2 5 8 19
3 4 9 18
12 11 10 17

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A081348, A080335.

[((1+2*n^2)+(1-2*n)*(-1)^n)/2: n in [0..50]]; // Vincenzo Librandi, Aug 08 2013

CoefficientList[Series[(1 + x - x^2 + 7 x^3) / ((1 - x)^3 (1 + x)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 08 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{1,2,3,12,13},60] (* Harvey P. Dale, Aug 13 2025 *)

a(n) = ((1+2*n^2)+(1-2*n)*(-1)^n)/2.
a(2n) = A054554(n).
a(2n+1) = 2*A000384(n).
G.f.: (1+x-x^2+7*x^3)/((1-x)^3*(1+x)^2). [Colin Barker, Apr 17 2012]

cp's OEIS Frontend

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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A073337 Primes of the form 4*k^2 - 10*k + 7 with k positive.

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A227776 a(n) = 6*n^2 + 1.

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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.

A073337 Primes of the form 4k^2 - 10k + 7 with k positive.