A080335 -id:A080335 - 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.

A137928 The even principal diagonal of a 2n X 2n square spiral.

Original entry on oeis.org

2, 4, 10, 16, 26, 36, 50, 64, 82, 100, 122, 144, 170, 196, 226, 256, 290, 324, 362, 400, 442, 484, 530, 576, 626, 676, 730, 784, 842, 900, 962, 1024, 1090, 1156, 1226, 1296, 1370, 1444, 1522, 1600, 1682, 1764, 1850, 1936, 2026, 2116, 2210, 2304, 2402, 2500, 2602, 2704, 2810

Offset: 1

William A. Tedeschi, Feb 29 2008

This is concerned with 2n X 2n square spirals of the form illustrated in the Example section.

			Example with n = 2:
.
   7---8---9--10
   |           |
   6   1---2  11
   |       |   |
   5---4---3  12
               |
  16--15--14--13
.
a(1) = 2(1) + 4*floor((1-1)/4) = 2;
a(2) = 2(2) + 4*floor((2-1)/4) = 4.

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

Cf. A000982, A002061 (odd diagonal), A002620, A080335, A171218.

A137928:=n->2*ceil(n^2/2): seq(A137928(n), n=1..100); # Wesley Ivan Hurt, Jul 25 2017

LinearRecurrence[{2,0,-2,1},{2,4,10,16},60] (* Harvey P. Dale, Aug 28 2017 *)

a(n)=2*n+(n-1)^2\4*4 \\ Charles R Greathouse IV, May 21 2015

a = lambda n: 2*n + 4*floor((n-1)**2/4)

a(n) = 2*n + 4*floor((n-1)^2/4) = 2*n + 4*A002620(n-1).
a(n) = A171218(n) - A171218(n-1). - Reinhard Zumkeller, Dec 05 2009
From R. J. Mathar, Jun 27 2011: (Start)
G.f.: 2*x*(1 + x^2) / ( (1 + x)*(1 - x)^3 ).
a(n) = 2*A000982(n). (End)
a(n+1) = (3 + 4*n + 2*n^2 + (-1)^n)/2 = A080335(n) + (-1)^n. - Philippe Deléham, Feb 17 2012
a(n) = 2 * ceiling(n^2/2). - Wesley Ivan Hurt, Jun 15 2013
a(n) = n^2 + (n mod 2). - Bruno Berselli, Oct 03 2017
Sum_{n>=1} 1/a(n) = Pi*tanh(Pi/2)/4 + Pi^2/24. - Amiram Eldar, Jul 07 2022

A081348 First row in maze arrangement of natural numbers.

Original entry on oeis.org

1, 6, 7, 20, 21, 42, 43, 72, 73, 110, 111, 156, 157, 210, 211, 272, 273, 342, 343, 420, 421, 506, 507, 600, 601, 702, 703, 812, 813, 930, 931, 1056, 1057, 1190, 1191, 1332, 1333, 1482, 1483, 1640, 1641, 1806, 1807, 1980, 1981, 2162, 2163, 2352, 2353, 2550

Offset: 0

Paul Barry, Mar 19 2003

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. A081347, A080335.

[(2*n^2+4*n+3-(2*n+1)*(-1)^n)/2: n in [0..50]]; // Vincenzo Librandi, Aug 08 2013

CoefficientList[Series[(1 + 5 x - x^2 + 3 x^3) / ((1 - x)^3 (1 + x)^2), {x, 0, 60}], x] (* Vincenzo Librandi, Aug 08 2013 *)

a(n) = (2*n^2+4*n+3-(2*n+1)(-1)^n)/2.
a(2*n) = A054569(n).
a(2*n+1) = 2*A014105(n+1).
G.f.: (1+5*x-x^2+3*x^3)/((1-x)^3*(1+x)^2). - Colin Barker, Apr 17 2012

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]

A086955 a(n) = n^2 + 2*n + 2 - (-1)^n.

Original entry on oeis.org

1, 6, 9, 18, 25, 38, 49, 66, 81, 102, 121, 146, 169, 198, 225, 258, 289, 326, 361, 402, 441, 486, 529, 578, 625, 678, 729, 786, 841, 902, 961, 1026, 1089, 1158, 1225, 1298, 1369, 1446, 1521, 1602, 1681, 1766, 1849, 1938, 2025, 2118, 2209, 2306, 2401, 2502

Offset: 0

Paul Barry, Jul 25 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. A080335.

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

A086955:=n->n^2+2*n+2-(-1)^n; seq(A086955(n), n=0..100); # Wesley Ivan Hurt, Nov 30 2013

Table[n^2+2n+2-(-1)^n, {n,0,100}] (* Wesley Ivan Hurt, Nov 30 2013 *)

G.f.: (1 + 4*x - 3*x^2 + 2*x^3)/((1+x)*(1-x)^3).

a(2*n) = (2*n+1)^2 = A016754(n), a(2*n+1) = 4*n^2 + 8*n + 6 = A005899(n+1).

A093650 Natural numbers arranged in a square maze beginning 1, 2, 3, then moving right, then up, right, down, left, down, right, etc., and read by antidiagonals upwards.

Original entry on oeis.org

1, 2, 6, 3, 5, 7, 12, 4, 8, 20, 13, 11, 9, 19, 21, 30, 14, 10, 18, 22, 42, 31, 29, 15, 17, 23, 41, 43, 56, 32, 28, 16, 24, 40, 44, 72, 57, 55, 33, 27, 25, 39, 45, 71, 73, 90, 58, 54, 34, 26, 38, 46, 70, 74, 110, 91, 89, 59, 53, 35, 37, 47, 69, 75, 109, 111

Offset: 1

Michael Joseph Halm, May 15 2004

			a(3) = 6 because the maze begins 2 under 1, 3 under 2, 4 right of 3, 5 right of 2 and 6 right of 1.
Array begins:
   1   6---7  20 ...
   |   |   |   |
   2   5   8  19 ...
   |   |   |   |
   3---4   9  18 ...
           |   |
  12--11--10  17 ...
   |           |
  13--14--15--16 ...
  ...

Cf. A080335, A081347, A081348.

Other square mazes: A081344, A081349.

More terms from Jinyuan Wang, Jun 15 2022

A357745 Numbers on the 8 main spokes of a square spiral with 1 in the center.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 19, 21, 23, 25, 28, 31, 34, 37, 40, 43, 46, 49, 53, 57, 61, 65, 69, 73, 77, 81, 86, 91, 96, 101, 106, 111, 116, 121, 127, 133, 139, 145, 151, 157, 163, 169, 176, 183, 190, 197, 204, 211, 218, 225, 233, 241, 249, 257, 265, 273

Offset: 1

Karl-Heinz Hofmann, Dec 22 2022

The 8 main spokes are (with 1 in the center, 2 to the east, 3 to the northeast): east: A054552; northeast: A054554; north: A054556; northwest: A053755; west: A054567; southwest: A054569; south: A033951; southeast: A016754.
Alternatively the 8 main spokes are pairwise part of the 4 main axes: horizontal: A317186; vertical: A267682; diagonal: A002061; antidiagonal: A080335.
And lastly the 4 main axes are giving two main crosses: Horizontal-vertical cross: A039823; Diagonal-antidiagonal cross: A200975.

			See visualization in links.

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000
Karl-Heinz Hofmann, Visualization of the first few terms
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,1,-2,1).

Cf. A054552, A054554, A054556, A053755, A054567, A054569, A033951, A016754.

Cf. A317186, A267682, A002061, A080335, A039823, A200975.

Rest@ CoefficientList[Series[x (1 - x^8 + x^9)/((1 - x)^3*(1 + x) (1 + x^2) (1 + x^4)), {x, 0, 63}], x] (* Michael De Vlieger, Dec 29 2022 *)
a[n_] := BitShiftRight[(n + 3)^2, 4] + Boole[BitAnd[n, 7] != 1]; Array[a, 65] (* Amiram Eldar, Dec 30 2022, after the PARI code *)
LinearRecurrence[{2,-1,0,0,0,0,0,1,-2,1},{1,2,3,4,5,6,7,8,9,11},70] (* Harvey P. Dale, Jul 13 2025 *)

a(n) = sqr(n+3)>>4 + (bitand(n,7)!=1); \\ Kevin Ryde, Dec 30 2022

def A357745(n): return ((n+3)**2 >> 4) + 1 if n % 8 != 1 else (n+3)**2 >> 4

G.f.: x*(1-x^8+x^9)/((1-x)^3*(1+x)*(1+x^2)*(1+x^4)). - Joerg Arndt, Dec 29 2022

a(n) = floor((n+3)^2 / 16) + (1 if n != 1 mod 8). - Kevin Ryde, Dec 30 2022

A087431 a(n) = 0^n/2 + 2^n*(n^2+n+2)/4.

Original entry on oeis.org

1, 2, 8, 28, 88, 256, 704, 1856, 4736, 11776, 28672, 68608, 161792, 376832, 868352, 1982464, 4489216, 10092544, 22544384, 50069504, 110624768, 243269632, 532676608, 1161822208, 2524971008, 5469372416, 11811160064, 25434259456

Offset: 0

Paul Barry, Sep 02 2003

Binomial transform of A080335 (with additional leading 1).

Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

LinearRecurrence[{6,-12,8},{1,2,8,28},30] (* Harvey P. Dale, Nov 26 2015 *)

a(n) = 2*A007466(n) for n >= 1.

G.f.: (1-4*x+8*x^2-4*x^3)/(1-2*x)^3. - Colin Barker, Mar 18 2012

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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A137928 The even principal diagonal of a 2n X 2n square spiral.

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Maple

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A081348 First row in maze arrangement of natural numbers.

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A081347 First column in maze arrangement of natural numbers.

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A086955 a(n) = n^2 + 2*n + 2 - (-1)^n.

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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.