A137932 -id:A137932 - cp's OEIS frontend

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

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.

A081353 Diagonal of square maze arrangement of natural numbers A081349.

Original entry on oeis.org

3, 5, 13, 19, 31, 41, 57, 71, 91, 109, 133, 155, 183, 209, 241, 271, 307, 341, 381, 419, 463, 505, 553, 599, 651, 701, 757, 811, 871, 929, 993, 1055, 1123, 1189, 1261, 1331, 1407, 1481, 1561, 1639, 1723, 1805, 1893, 1979, 2071, 2161, 2257, 2351, 2451, 2549

Offset: 0

Paul Barry, Mar 19 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A081349, A081350, A081351, A081352.
Cf. A109613, A137932.
Bisections are in A054554, A125202.

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

A081353:=n->(n+1)*(n+2)+(-1)^n: seq(A081353(n), n=0..100); # Wesley Ivan Hurt, Aug 09 2015

Table[(n + 1) (n + 2) + (-1)^n, {n, 0, 70}] (* Wesley Ivan Hurt, Aug 09 2015 *)
LinearRecurrence[{2,0,-2,1},{3,5,13,19},50] (* Harvey P. Dale, Aug 02 2021 *)

a(n) = (n+1)*(n+2)+(-1)^n = 2*binomial(n+2,2)+(-1)^n.
G.f.: (3-x)*(1+x^2)/((1-x)^3*(1+x)). [Colin Barker, Sep 03 2012]
From Wesley Ivan Hurt, Aug 09 2015: (Start)
a(n) = 2*a(n-1)-2*a(n-3)+a(n-4), n>4.
a(n) = n^2+3n+3 if n is even, otherwise n^2+3n+1.
a(n) = A137932(n+3) - A109613(n+1). (End)

A213041 Number of triples (w,x,y) with all terms in {0..n} and 2*|w-x| = max(w,x,y) - min(w,x,y).

Original entry on oeis.org

1, 2, 7, 12, 21, 30, 43, 56, 73, 90, 111, 132, 157, 182, 211, 240, 273, 306, 343, 380, 421, 462, 507, 552, 601, 650, 703, 756, 813, 870, 931, 992, 1057, 1122, 1191, 1260, 1333, 1406, 1483, 1560, 1641, 1722, 1807, 1892, 1981, 2070, 2163, 2256

Offset: 0

Clark Kimberling, Jun 10 2012

See A212959 for a guide to related sequences.

For n > 3, a(n-2) is the number of distinct values of the magic constant in a perimeter-magic (n-1)-gon of order n (see A342819). - Stefano Spezia, Mar 23 2021

Terrel Trotter, Perimeter-Magic Polygons, Journal of Recreational Mathematics Vol. 7, No. 1, 1974, pp. 14-20 (see equations 10-13).
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A002620, A004526, A058331, A212959, A168277 (first differences), A342819.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Max[w, x, y] - Min[w, x, y] == 2 Abs[w - x],
s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 45]]   (* A213041 *)

Vec((1+3*x^2)/((1-x)^3*(1+x)) + O(x^99)) \\ Altug Alkan, May 06 2016

a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n > 3.
G.f.: (1 + 3*x^2)/((1 - x)^3 * (1 + x)).
a(n) = (n+1)^2 - 2*A004526(n-1) - 2. - Wesley Ivan Hurt, Jul 15 2013
a(n) = A002620(n+2) + 3*A002620(n). - R. J. Mathar, Jul 15 2013
a(n)+a(n+1) = A058331(n+1). - R. J. Mathar, Jul 15 2013
a(n) = n*(n+1) + (1+(-1)^n)/2. - Wesley Ivan Hurt, May 06 2016
E.g.f.: x*(x + 2)*exp(x) + cosh(x). - Ilya Gutkovskiy, May 06 2016
a(n) = A000384(n+1) - A137932(n+2). - Federico Provvedi, Aug 17 2023

A267089 T(n,k) is decimal conversion of 1's in an n X n table that lie on its principal diagonals.

Original entry on oeis.org

1, 3, 3, 5, 2, 5, 9, 6, 6, 9, 17, 10, 4, 10, 17, 33, 18, 12, 12, 18, 33, 65, 34, 20, 8, 20, 34, 65, 129, 66, 36, 24, 24, 36, 66, 129, 257, 130, 68, 40, 16, 40, 68, 130, 257, 513, 258, 132, 72, 48, 48, 72, 132, 258, 513, 1025, 514, 260, 136, 80, 32, 80, 136, 260, 514

Offset: 0

Kival Ngaokrajang, Jan 10 2016

Inspired by A137932 and A042948.
Conjectures:
(i) The first column is A083318.
(ii) T(n,k) = A086066(m) where m >= 10, n = m - 9*k, k = floor(m/10).

			See the "Illustration of initial terms" link for explicit examples.
Triangle begins:
n\k 0   1  2  3  4  5  6   7   8 ...
0   1
1   3   3
2   5   2  5
3   9   6  6  9
4  17  10  4 10 17
5  33  18 12 12 18 33
6  65  34 20  8 20 34 65
7 129  66 36 24 24 36 66 129
8 257 130 68 40 16 40 68 130 257
...

Kival Ngaokrajang, Illustration of initial terms, T(n,k) for n = 0..15, k = 0..10

Cf. A042948, A083318, A086066, A137932.

A267489 a(n) = n^2 - 4*floor(n^2/6).

Original entry on oeis.org

0, 1, 4, 5, 8, 9, 12, 17, 24, 29, 36, 41, 48, 57, 68, 77, 88, 97, 108, 121, 136, 149, 164, 177, 192, 209, 228, 245, 264, 281, 300, 321, 344, 365, 388, 409, 432, 457, 484, 509, 536, 561, 588, 617, 648, 677, 708, 737, 768, 801, 836, 869, 904

Offset: 0

Kival Ngaokrajang, Jan 16 2016

Inspired by A137932 and A042948.
The pattern is generated by adding subdiagonals parallel to principal diagonals at a spacing of at least 1 box in any direction from the previous generation.
Conjectures:
(i) a(n) is the total number of boxes (or 1's) at the n-th iteration.
(ii) The total number of left boxes (or 0's) is 4*A056827.

Colin Barker, Table of n, a(n) for n = 0..1000
Kival Ngaokrajang, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,1,-2,1).

Cf. A042948, A056827, A137932.

[0] cat [n^2-4*Floor(n^2/6): n in [1..70]]; // Vincenzo Librandi, Jan 16 2016

A267489:=n->n^2-4*floor(n^2/6): seq(A267489(n), n=0..100); # Wesley Ivan Hurt, Apr 11 2017

Table[n^2 - 4 Floor[n^2 / 6], {n, 0, 70}] (* Vincenzo Librandi, Jan 16 2016 *)

for (n = 0, 100, a = n^2-4*floor(n^2/6); print1(a, ", "))

concat(0, Vec(x*(1+2*x-2*x^2+2*x^3-2*x^4+2*x^5+x^6)/((1-x)^3*(1+x)*(1-x+x^2)*(1+x+x^2)) + O(x^100))) \\ Colin Barker, Jan 16 2016

a(n)=n^2 - n^2\6*4 \\ Charles R Greathouse IV, Mar 22 2017

a(n) = n^2 - 4*floor(n^2/6) for n >= 0.
From Colin Barker, Jan 16 2016: (Start)
a(n) = 2*a(n-1) - a(n-2) + a(n-6) - 2*a(n-7) + a(n-8) for n>7.
G.f.: x*(1+2*x-2*x^2+2*x^3-2*x^4+2*x^5+x^6) / ((1-x)^3*(1+x)*(1-x+x^2)*(1+x+x^2)).
(End)

A279403 Irregular triangle read by rows: T(n,k) (n>=1, 0 <= k <= n^2) = minimal number of squares not attacked by k queens on an n X n toroidal board, with trailing zeros truncated.

Original entry on oeis.org

1, 4, 9, 16, 4, 25, 8, 2, 36, 16, 4, 49, 24, 11, 4, 64, 36, 16, 6, 81, 48, 27, 12, 3, 100, 64, 36, 19, 4, 121, 80, 51, 29, 13, 144, 100, 64, 39, 16, 6, 169, 120, 83, 53, 29, 8, 2, 196, 144, 100, 67, 36, 18, 8, 225, 168, 223, 82, 41, 256, 196, 144, 103, 64, 40

Offset: 1

Andrey Zabolotskiy, Dec 11 2016

Row lengths are A279402.

			The triangle begins:
1 (0)
4 (0, 0, 0, 0)
9 (0, 0, ...)
16 4 (0, 0, ...)
25 8 2
36 16 4
49 24 11 4
64 36 16 6
81 48 27 12 3
100 64 36 19 4
121 80 51 29 13
144 100 64 39 16 6
169 120 83 53 29 8 2
196 144 100 67 36 18 8
225 168 123 82 41
256 196 144 103 64 40 ...

Cf. A279402, A279406.

T(n,0) = A000290(n).
T(n,1) = A000290(n)-A047461(n) = A137932(n-1).
T(n,2) = A248825(n-4) for n >= 6.

A267090 Triangle read by rows: Fill an n X n square with 1's, except for 0's on the two main diagonals. Then T(n,k) is decimal equivalent of the k-th row (0<=k<=n).

Original entry on oeis.org

0, 0, 0, 2, 5, 2, 6, 9, 9, 6, 14, 21, 27, 21, 14, 30, 45, 51, 51, 45, 30, 62, 93, 107, 119, 107, 93, 62, 126, 189, 219, 231, 231, 219, 189, 126, 254, 381, 443, 471, 495, 471, 443, 381, 254, 510, 765, 891, 951, 975, 975, 951, 891, 765, 510

Offset: 0

Kival Ngaokrajang, Jan 10 2016

Inspired by A137932 and A042948.
Conjectures:
(i) The first column is A000225/2.
(ii) For even-n, T(n,n/2) = A129868.
(iii) For odd-n, T(n,(n-1)/2) = T(n,(n+1)/2) = A220236.

			Triangle begins:
n\k  0   1   2   3   4   5   6   7   8 ...
0    0
1    0   0
2    2   5   2
3    6   9   9   6
4   14  21  27  21  14
5   30  45  51  51  45  30
6   62  93 107 119 107  93  62
7  126 189 219 231 231 219 189 126
8  254 381 443 471 495 471 443 381 254
...

Kival Ngaokrajang, Illustration of initial terms, T(n,k) for n = 0..10, k = 0..10

Cf. A000225, A042948, A129868, A137932, A267089.

A301648 Number of longest cycles in the n X n grid graph.

Original entry on oeis.org

0, 1, 5, 6, 226, 1072, 255088, 4638576, 6663430912, 467260456608, 3916162476483538, 1076226888605605706, 51249820944023435573470, 56126499620491437281263608, 14870957102232406137455708164254, 65882516522625836326159786165530572, 95494789899510664733921727510895952184006

Offset: 1

Eric W. Weisstein, Mar 25 2018

nonn

a(10) = 467260456608.

Eric Weisstein's World of Mathematics, Graph Cycle
Eric Weisstein's World of Mathematics, Grid Graph

Cf. A137932 (circumference of the (n-1) X (n-1) grid graph).
Cf. A003763 (number of Hamiltonian cycles in the 2n X 2n grid graph).
Cf. A181584 (number of longest cycles in the (2n+1) X (2n+1) grid graph).

a(2n) = A003763(n).

a(2n+1) = A181584(n). - Andrew Howroyd, Mar 01 2020

Terms a(9) and beyond from Andrew Howroyd, Mar 01 2020

cp's OEIS Frontend

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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A081353 Diagonal of square maze arrangement of natural numbers A081349.

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A213041 Number of triples (w,x,y) with all terms in {0..n} and 2*|w-x| = max(w,x,y) - min(w,x,y).

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A267089 T(n,k) is decimal conversion of 1's in an n X n table that lie on its principal diagonals.

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A267489 a(n) = n^2 - 4*floor(n^2/6).

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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.