A054552 -id:A054552 - cp's OEIS frontend

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

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.

A244677 The spiral of Champernowne, read along the East ray.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 04 2014

Inspired by Stanislaw Ulam's spiral, circa 1963.

			The beginning of the infinite spiral of David Gawen Champernowne:
.
  7--1--9--6--1--8--6--1--7--6--1--6--6--1--5--6--1--4--6--1--3  .
  |                                                           |  |
  0  1--4--4--1--3--4--1--2--4--1--1--4--1--0--4--1--9--3--1  6  .
  |  |                                                     |  |  |
  1  4  2--1--1--2--1--0--2--1--9--1--1--8--1--1--7--1--1  8  1  .
  |  |  |                                               |  |  |  |
  7  5  2  0--1--1--0--1--0--0--1--9--9--8--9--7--9--6  6  3  2  9
  |  |  |  |                                         |  |  |  |  |
  1  1  1  2  7--7--6--7--5--7--4--7--3--7--2--7--1  9  1  1  6  8
  |  |  |  |  |                                   |  |  |  |  |  |
  1  4  2  1  7  5--5--4--5--3--5--2--5--1--5--0  7  5  1  7  1  1
  |  |  |  |  |  |                             |  |  |  |  |  |  |
  7  6  3  0  8  5  7--3--6--3--5--3--4--3--3  5  0  9  5  3  1  8
  |  |  |  |  |  |  |                       |  |  |  |  |  |  |  |
  2  1  1  3  7  6  3  3--2--2--2--1--2--0  3  9  7  4  1  1  6  8
  |  |  |  |  |  |  |  |                 |  |  |  |  |  |  |  |  |
  1  4  2  1  9  5  8  2  3--1--2--1--1  2  2  4  9  9  1  6  1  1
  |  |  |  |  |  |  |  |  |           |  |  |  |  |  |  |  |  |  |
  7  7  4  0  8  7  3  4  1  5--4--3  1  9  3  8  6  3  4  3  0  7
  |  |  |  |  |  |  |  |  |  |     |  |  |  |  |  |  |  |  |  |  |
  3  1  1  4  0  5  9  2  4  6  1--2  0  1  1  4  8  9  1  1  6  8
  |  |  |  |  |  |  |  |  |  |        |  |  |  |  |  |  |  |  |  |
  1  4  2  1  8  8  4  5  1  7--8--9--1  8  3  7  6  2  1  5  1  1
  |  |  |  |  |  |  |  |  |              |  |  |  |  |  |  |  |  |
  7  8  5  0  1  5  0  2  5--1--6--1--7--1  0  4  7  9  3  3  9  6
  |  |  |  |  |  |  |  |                    |  |  |  |  |  |  |  |
  4  1  1  5  8  9  4  6--2--7--2--8--2--9--3  6  6  1  1  1  5  8
  |  |  |  |  |  |  |                          |  |  |  |  |  |  |
  1  4  2  1  2  6  1--4--2--4--3--4--4--4--5--4  6  9  1  4  1  1
  |  |  |  |  |  |                                |  |  |  |  |  |
  7  9  6  0  8  0--6--1--6--2--6--3--6--4--6--5--6  0  2  3  8  5
  |  |  |  |  |                                      |  |  |  |  |
  5  1  1  6  3--8--4--8--5--8--6--8--7--8--8--8--9--9  1  1  5  8
  |  |  |  |                                            |  |  |  |
  1  5  2  1--0--7--1--0--8--1--0--9--1--1--0--1--1--1--1  3  1  1
  |  |  |                                                  |  |  |
  7  0  7--1--2--8--1--2--9--1--3--0--1--3--1--1--3--2--1--3  7  4
  |  |                                                        |  |
  6  1--5--1--1--5--2--1--5--3--1--5--4--1--5--5--1--5--6--1--5  8
  |                                                              |
  1--7--7--1--7--8--1--7--9--1--8--0--1--8--1--1--8--2--1--8--3--1

Robert G. Wilson v, Cover of the March 1964 issue of Scientific American

Cf. A033307, A054552, A244678 - A244688, A033952, A244690 - A244692.

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 - 11n + 8 (* see formula section *); Array[ almostNatural[ f@#, 10] &, 105]

Formulas for rays in directions of 32 compass points:
  SE     4n^2   -4n  +1
  SExS  64n^2 -113n +50
  SSE   16n^2  -25n +10
  SxE   64n^2 -115n +52
  S      4n^2   -5n  +2
  SxW   64n^2 -117n +54
  SSW   16n^2  -27n +12
  SWxS  64n^2 -119n +56
  SW     4n^2   -6n  +3
  SWxW  64n^2 -121n +58
  WSW   16n^2  -29n +14
  WxS   64n^2 -123n +60
  W      4n^2   -7n  +4
  WxN   64n^2 -125n +62
  WNW   16n^2  -31n +16
  NWxW  64n^2 -127n +64
  NW     4n^2   -8n  +5
  NWxN  64n^2 -129n +66
  NNW   16n^2  -33n +18
  NxW   64n^2 -131n +68
  N      4n^2   -9n  +6
  NxE   64n^2 -133n +70
  NNE   16n^2  -35n +20
  NExN  64n^2 -135n +72
  NE     4n^2  -10n  +7
  NExE  64n^2 -137n +74
  ENE   16n^2  -37n +22
  ExN   64n^2 -139n +76
  E      4n^2  -11n  +8
  ExS   64n^2 -141n +78
  ESE   16n^2  -39n +24
  SExE  64n^2 -143n +80

A068225 Neighbor in 1-2 direction of numbers arranged as clockwise spiral.

Original entry on oeis.org

2, 11, 12, 3, 4, 1, 8, 9, 10, 27, 28, 29, 30, 13, 14, 15, 16, 5, 6, 7, 22, 23, 24, 25, 26, 51, 52, 53, 54, 55, 56, 31, 32, 33, 34, 35, 36, 17, 18, 19, 20, 21, 44, 45, 46, 47, 48, 49, 50, 83, 84, 85, 86, 87, 88, 89, 90, 57, 58, 59, 60, 61, 62, 63, 64, 37, 38, 39, 40, 41, 42, 43

Offset: 1

Frank Ellermann, Feb 22 2002

			For example, if the spiral is oriented such that 1 is immediately to the right of 2, as shown below, then a(2) = 11 because 11 is immediately to the right of 2.
  21--22--23--24--25--26
  |                    |
  20  7---8---9---10  27
  |   |            |   |
  19  6   1---2   11  28
  |   |       |    |   |
  18  5---4---3   12  29
  |                |   |
  17--16--15--14--13  30
                       |
  36--35--34--33--32--31

Peter Kagey, Table of n, a(n) for n = 1..10000
Frank Ellermann, Some EIS sequences based on a spiral of natural numbers.
Index entries for sequences that are permutations of the natural numbers.

Cf. A068226 (inverse, left), A334751 (above), A334752 (below).

Cf. A054552 (middle row right), A054567 (middle row left).

A054925 a(n) = ceiling(n*(n-1)/4).

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 8, 11, 14, 18, 23, 28, 33, 39, 46, 53, 60, 68, 77, 86, 95, 105, 116, 127, 138, 150, 163, 176, 189, 203, 218, 233, 248, 264, 281, 298, 315, 333, 352, 371, 390, 410, 431, 452, 473, 495, 518, 541, 564, 588, 613, 638, 663, 689, 716, 743, 770, 798

Offset: 0

N. J. A. Sloane, May 24 2000

Number of edges in "median" graph - gives positions of largest entries in rows of table in A054924.
Form the clockwise spiral starting 0,1,2,....; then A054925(n+1) interleaves 2 horizontal (A033951, A033991) and 2 vertical (A007742, A054552) branches. A bisection is A014848. - Paul Barry, Oct 08 2007
Consider the standard 4-dimensional Euclidean lattice. We take 1 step along the positive x-axis, 2 along the positive y-axis, 3 along the positive z-axis, 4 along the positive t-axis, and then back round to the x-axis. This sequence gives the floor of the Euclidean distance to the origin after n steps. - Jon Perry, Apr 16 2013
Jon Perry's JavaScript code is explained by A238604. - Michael Somos, Mar 01 2014
Ceiling of the area under the polygon connecting the lattice points (n, floor(n/2)) from 0..n. - Wesley Ivan Hurt, Jun 09 2014
Ceiling of one-half of each triangular number. - Harvey P. Dale, Oct 03 2016
For n > 2, also the edge cover number of the (n-1)-triangular honeycomb queen graph. - Eric W. Weisstein, Jul 14 2017
Conjecture: For n>11, there always exists a prime number p such that a(n)Raul Prisacariu, Sep 01 2024
For n = 1 up to at least n = 13, also the lower matching number of the triangular honeycomb bishop graph. - Eric W. Weisstein, Dec 13 2024
Conjecturally, apart from the first term, the sequence terms are the exponents in the expansion of Sum_{n >= 0} q^(3*n) * (Product_{k >= 2*n+1} 1 - q^k) = 1 - q - q^2 + q^3 + q^5 - q^8 - q^11 + + - - .... Cf. A039825. - Peter Bala, Apr 13 2025

			a(6) = 8; ceiling(6*(6-1)/4) = ceiling(30/4) = 8.
G.f. = x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 8*x^6 + 11*x^7 + 14*x^8 + 18*x^9 + 23*x^10 + ...

Ivan Panchenko, Table of n, a(n) for n = 0..10000
Craig Knecht, Binary fill patterns in a triangle refilled with natural numbers.
Eric Weisstein's World of Mathematics, Edge Cover Number.
Eric Weisstein's World of Mathematics, Triangular Honeycomb Bishop Graph.
Eric Weisstein's World of Mathematics, Triangular Honeycomb Queen Graph.
Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).

Cf. A007742, A033951, A033991, A039825, A054552, A054924, A054925 + A011848 = C(n, 2).

Cf. A213172, A238604.

p=new Array(0,0,0,0);
for (a=0;a<100;a++) {
p[a%4]+=a;
document.write(Math.floor(Math.sqrt(p[0]*p[0]+p[1]*p[1]+p[2]*p[2]+p[3]*p[3]))+", ");
} /* Jon Perry, Apr 16 2013 */

[ Ceiling(n*(n-1)/4) : n in [0..50] ]; // Wesley Ivan Hurt, Jun 09 2014

I:=[0,0,1,2,3]; [n le 5 select I[n] else 3*Self(n-1)-4*Self(n-2)+4*Self(n-3)-3*Self(n-4)+Self(n-5): n in [1..60]]; // Vincenzo Librandi, Jul 14 2015

seq(ceil(binomial(n,2)/2), n=0..57); # Zerinvary Lajos, Jan 12 2009

Table[Ceiling[(n^2 - n)/4], {n, 0, 20}] (* Wesley Ivan Hurt, Nov 01 2013 *)
LinearRecurrence[{3, -4, 4, -3, 1}, {0, 0, 1, 2, 3}, 60] (* Vincenzo Librandi, Jul 14 2015 *)
Join[{0}, Ceiling[#/2] &/ @ Accumulate[Range[0, 60]]] (* Harvey P. Dale, Oct 03 2016 *)
Ceiling[Binomial[Range[0, 20], 2]/2] (* Eric W. Weisstein, Dec 13 2024 *)
Table[Ceiling[Binomial[n, 2]/2], {n, 0, 20}] (* Eric W. Weisstein, Dec 13 2024 *)
Table[(1 + (n - 1) n - Cos[n Pi/2] - Sin[n Pi/2])/4, {n, 0, 20}] (* Eric W. Weisstein, Dec 13 2024 *)
CoefficientList[Series[x^2 (-1 + x - x^2)/((-1 + x)^3 (1 + x^2)), {x, 0, 20}], x] (* Eric W. Weisstein, Dec 13 2024 *)

{a(n) = ceil( n * (n-1)/4)}; /* Michael Somos, Feb 11 2004 */

[ceil(binomial(n,2)/2) for n in range(0,58)] # Zerinvary Lajos, Dec 01 2009

Euler transform of length 6 sequence [ 2, 0, 1, 1, 0, -1]. - Michael Somos, Sep 02 2006
From Michael Somos, Feb 11 2004: (Start)
G.f.: x^2 * (x^2 - x + 1) / ((1 - x)^3 * (1 + x^2)) = x^2 * (1 - x^6) / ((1 - x)^2 * (1 - x^3) * (1 - x^4)).
a(1-n) = a(n).
A011848(n) = a(-n). (End)
From Michael Somos, Mar 01 2014: (Start)
a(n + 4) = a(n) + 2*n + 3.
a(n+1) = floor( sqrt( A238604(n))). (End)
a(n) = A011848(n) + A133872(n+2). - Wesley Ivan Hurt, Jun 09 2014
Sum_{n>=2} 1/a(n) = 4 - Pi + 2*Pi*sinh(sqrt(7)*Pi/4)/(sqrt(7)*(1/sqrt(2)+cosh(sqrt(7)*Pi/4))). - Amiram Eldar, Dec 23 2024

A244807 The hexagonal spiral of Champernowne, read along the East (or 90-degree) ray.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 06 2014

Inspired by Stanislaw M. Ulam's hexagonal spiral, circa 1963. See example section of A056105.

When A056105, A056106, A056107, A056108, A056109 & A003215 were submitted, the offsets were 0. Here the offset is 1.

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

Cf. A007376, A054552, A056105, A244808, A244809, A244810, A244811, A244812, A244813, A244814, A244815, A244816, A244817, A244818.

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_] := 3n^2- 8n +6 (* see formula section of A244807 *); Array[ almostNatural[ f@#, 10] &, 105]

For each 30 degrees of the compass, the corresponding spoke (or ray) has a generating formula as follows:
 3n^2- 8n +6
12n^2-27n+16
 3n^2- 7n+ 5
12n^2-25n+14
 3n^2 -6n +4
12n^2-23n+12
 3n^2 -5n +3
12n^2-21n+10
 3n^2 -4n +2
12n^2-19n +8
 3n^2 -3n +1
12n^2-17n+ 6
Also see formula section of A056105.

A054551 Prime number spiral (clockwise, North spoke).

Original entry on oeis.org

2, 3, 31, 107, 241, 443, 709, 1049, 1471, 1973, 2539, 3191, 3911, 4729, 5651, 6637, 7699, 8867, 10133, 11503, 12941, 14537, 16073, 17863, 19727, 21601, 23609, 25759, 27967, 30319, 32719, 35201, 37831, 40627, 43391, 46399, 49411, 52553, 55813

Offset: 0

Enoch Haga and G. L. Honaker, Jr. Apr 09 2000

Smallest prime in n-th shell of prime spiral.
8-spoke wheel overlays prime number spiral; hub is 2 in shell 0; 8 spokes radiate from this hub; this is North, clockwise.
Shell 1 comprises the primes 3 5 7 11 13 17 19 23; 3 is lowest, 23 is highest.
The wheel may be rotated, but the sequences though pointing in different directions, will remain the same.

			Begin a prime number spiral at zero, proceed clockwise, North.
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
Hermetic Systems, Prime Number Spiral.

Cf. A000040, A054552, A054553, A054555, A054564, A054566, A054568, A054570, A053999.

Table[ Prime[4n^2 - 3n + 1], {n, 0, 40} ]

a(n) = A000040(A054552(n)). - R. J. Mathar, Aug 29 2018

Edited by Robert G. Wilson v, Feb 25 2002

A168022 Noncomposite numbers in the eastern ray of the Ulam spiral as oriented on the March 1964 cover of Scientific American.

Original entry on oeis.org

1, 2, 11, 53, 127, 233, 541, 743, 977, 1871, 3511, 4001, 4523, 5077, 9851, 11503, 12377, 14221, 16193, 19391, 20521, 21683, 22877, 24103, 29327, 30713, 33581, 42953, 55343, 57241, 63127, 67211, 80231, 84827, 91961, 101921, 104491, 123377

Offset: 0

Alonso del Arte, Nov 16 2009

Although 1 was not considered a prime number in Ulam's time, the March 1964 cover of Scientific American shows 1 highlighted in the same way as the primes.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Alonso del Arte, Ulam spiral (2009). [Note that "East" and "West" in this video match the cover of Scientific American, but "North" and "South" are switched.]
BackIssues.com, Scientific American March 1964 back issue
MathWorld, Prime Spiral
Scientific American, March 1964 cover
Wikipedia, Ulam Spiral

Cf. A054552, all numbers of the form 4n^2 - 3n + 1. Primes of northeastern ray are in A073337. Noncomposites of the northern ray are in A168023. Noncomposites of the northwestern ray are in A168024. Noncomposites of the western ray are in A168025. Noncomposites of the southwestern ray are in A168026. Noncomposites of the southern ray are in A168027.

Select[Table[4 n^2 - 3 n + 1, {n, 0, 199}], Length[Divisors[ # ]] < 3 &]

Positive numbers of the form 4n^2 - 3n + 1 with no more than two divisors.

A033293 A Connell-like sequence: take 1 number = 1 (mod Q), 2 numbers = 2 (mod Q), 3 numbers = 3 (mod Q), etc., where Q = 8.

Original entry on oeis.org

1, 2, 10, 11, 19, 27, 28, 36, 44, 52, 53, 61, 69, 77, 85, 86, 94, 102, 110, 118, 126, 127, 135, 143, 151, 159, 167, 175, 176, 184, 192, 200, 208, 216, 224, 232, 233, 241, 249, 257, 265, 273, 281, 289, 297, 298, 306, 314, 322, 330, 338, 346, 354, 362, 370, 371, 379, 387, 395, 403, 411

Offset: 1

Gary E. Stevens

Reinhard Zumkeller, Rows n=1..120 of triangle, flattened
Gary E. Stevens, A Connell-Like Sequence, J. Integer Sequences, 1 (1998), #98.1.4.

Cf. A054552 (left edge), A001107 (right edge), A204674 (row sums), A204675 (central terms).

a033293 n k = a033293_tabl !! (n-1) !! (k-1)
a033293_row n = a033293_tabl !! (n-1)
a033293_tabl = f 1 [1..] where
   f k xs = ys : f (k+1) (dropWhile (<= last ys) xs) where
     ys  = take k $ filter ((== 0) . (`mod` 8) . (subtract k)) xs
-- Reinhard Zumkeller, Jan 18 2012 2011

row[1] = {1}; row[n_] := row[n] = Table[row[n-1][[-1]] + 8k + 1, {k, 0, n-1}]; Table[row[n], {n, 1, 11}] // Flatten (* Jean-François Alcover, Jan 25 2013 *)

More terms from jeroen.lahousse(AT)icl.com

Offset changed by Reinhard Zumkeller, Jan 18 2012

A214226 Sum of the eight nearest neighbors of n in a triangular horizontal-last spiral with positive integers.

Original entry on oeis.org

72, 80, 60, 76, 132, 100, 164, 112, 136, 200, 144, 128, 128, 136, 156, 196, 284, 220, 204, 208, 324, 224, 232, 248, 288, 384, 296, 264, 256, 264, 272, 280, 288, 296, 324, 380, 500, 404, 372, 368, 376, 384, 548, 400, 408, 416, 424, 448, 504, 632, 512, 464, 448

Offset: 1

Alex Ratushnyak, Jul 07 2012

			Triangular spiral begins:
__ __ __ __ __ __ __ 43
__ __ __ __ __ __ 42 21 44
__ __ __ __ __ 41 20  7 22 45
__ __ __ __ 40 19  6  1  8 23 46
__ __ __ 39 18  5  4  3  2  9 24 47
__ __ 38 17 16 15 14 13 12 11 10 25 48
__ 37 36 35 34 33 32 31 30 29 28 27 26 49
64 63 62 61 60 69 58 57 56 55 54 53 52 51 50
The eight nearest neighbors of 3 are 6, 1, 8, 4, 2, 14, 13, 12; their sum is a(3)=60.

Cf. A002061 - numbers on the central vertical axis.
Cf. A054552 - numbers on the axis starting with 1 and 2.
Cf. A214227 - sum of the four nearest neighbors.
Cf. A214250 - same sum in a triangular "horizontal-first" spiral.

SIZE=27 # must be odd
grid = [0] * (SIZE*SIZE)
saveX = [0]* (SIZE*SIZE)
saveY = [0]* (SIZE*SIZE)
saveX[1] = saveY[1] = posX = posY = SIZE//2
grid[posY*SIZE+posX]=1
n = 2
def walk(stepX,stepY,chkX,chkY,chkX2,chkY2):
  global posX, posY, n
  while 1:
    posX+=stepX
    posY+=stepY
    grid[posY*SIZE+posX]=n
    saveX[n]=posX
    saveY[n]=posY
    n+=1
    if posX==0 or grid[(posY+chkY)*SIZE+posX+chkX]+grid[(posY+chkY2)*SIZE+posX+chkX2]==0:
        return
while 1:
    walk( 1, 1, -1,  0, -1, 0)    # right+down
    walk(-1, 0,  1, -1, 0, -1)    # left
    if posX==0:
        break
    walk( 1,-1,  1, 1, 1, 1)    # right+up
for n in range(1,101):
    posX = saveX[n]
    posY = saveY[n]
    k = grid[(posY-1)*SIZE+posX] + grid[(posY+1)*SIZE+posX]
    k+= grid[(posY-1)*SIZE+posX-1] + grid[(posY-1)*SIZE+posX+1]
    k+= grid[(posY+1)*SIZE+posX-1] + grid[(posY+1)*SIZE+posX+1]
    k+= grid[posY*SIZE+posX-1] + grid[posY*SIZE+posX+1]
    print(k, end=', ')

A185438 a(n) = 8n^2 - 2n + 1.

Original entry on oeis.org

1, 7, 29, 67, 121, 191, 277, 379, 497, 631, 781, 947, 1129, 1327, 1541, 1771, 2017, 2279, 2557, 2851, 3161, 3487, 3829, 4187, 4561, 4951, 5357, 5779, 6217, 6671, 7141, 7627, 8129, 8647, 9181, 9731, 10297, 10879, 11477, 12091, 12721, 13367, 14029, 14707, 15401, 16111, 16837, 17579

Offset: 0

Paul Curtz, Feb 03 2011

Odd numbers (A005408) written clockwise as a square spiral:
.
  41--43--45--47--49--51
   |                   |
  39  13--15--17--19  53
   |   |           |   |
  37  11   1---3  21  55
   |   |       |   |   |
  35   9---7---5  23  57
   |               |   |
  33--31--29--27--25  59
                       |
  71--69--67--65--63--61
.
Walking in straight lines away from the center:
  1, 17, 49, ... = A069129(n+1) =  1 - 8*n + 8*n^2,
  1,  3, 21, ... = A033567(n)   =  1 - 6*n + 8*n^2,
  1, 15, 45, ... = A014634(n)   =  1 + 6*n + 8*n^2,
  1,  5, 25, ... = A080856(n)   =  1 - 4*n + 8*n^2,
  1, 13, 41, ... = A102083(n)   =  1 + 4*n + 8*n^2,
  1,  7, 29, ... = a(n)         =  1 - 2*n + 8*n^2,
  1, 11, 37, ... = A188135(n)   =  1 + 2*n + 8*n^2,
  1,  9, 33, ... = A081585(n)   =  1       + 8*n^2,
  5, 29, 69, ... = A108928(n+1) = -3       + 8*n^2,
  7, 31, 71, ... = A157914(n+1) = -1       + 8*n^2,
  9, 35, 77, ... = A033566(n+1) = -1 + 2*n + 8*n^2.
All are quadrisections of sequences in A181407(n) (example: A014634(n) and A033567(n) in A064038(n+1)) or of this family (?): a(n) is a quadrisection of f(n) = 1,1,1,1,2,7,11,8,11,29,37,23,28,67,79,46,... f(n) is just before A064038(n+1) (fifth vertical) in A181407(n). The companion to a(n) is A188135(n), another quadrisection of f(n). Two last quadrisections of f(n) are A054552(n) and A033951(n).
For n >= 1, bisection of A193867. - Omar E. Pol, Aug 16 2011
Also the sequence may be obtained by starting with the segment (1, 7) followed by the line from 7 in the direction 7, 29, ... in the square spiral whose vertices are the generalized hexagonal numbers (A000217). - Omar E. Pol, Aug 01 2016

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A005408, A014634, A014635, A033566, A033567, A033951, A054552, A064038.

Cf. A069129, A080856, A081585, A102083, A108928, A157914, A181407, A188135, A193867.

[1-2*n+8*n^2: n in [0..50]]; // Vincenzo Librandi, Feb 03 2011

Table[1 - 2n + 8n^2, {n, 0, 39}] (* Alonso del Arte, Feb 03 2011 *)
CoefficientList[Series[(-1 - 4 x - 11 x^2)/(x - 1)^3, {x, 0, 47}], x] (* Michael De Vlieger, Aug 01 2016 *)

a(n)=8*n^2-2*n+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = a(n-1) + 16*n - 10 (n > 0).
a(n) = 2*a(n-1) - a(n-2) + 16 (n > 1).
a(n) = 3*(n-1) - 3*a(n-2) + a(n-3) (n > 2).
G.f.: (-1 - 4*x - 11*x^2)/(x-1)^3. - R. J. Mathar, Feb 03 2011
a(n) = A014635(n) + 1. - Bruno Berselli, Apr 09 2011
E.g.f.: exp(x)*(1 + 6*x + 8*x^2). - Elmo R. Oliveira, Nov 17 2024

cp's OEIS Frontend

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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A244677 The spiral of Champernowne, read along the East ray.

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A068225 Neighbor in 1-2 direction of numbers arranged as clockwise spiral.

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A054925 a(n) = ceiling(n*(n-1)/4).

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A244807 The hexagonal spiral of Champernowne, read along the East (or 90-degree) ray.

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A054551 Prime number spiral (clockwise, North spoke).

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A168022 Noncomposite numbers in the eastern ray of the Ulam spiral as oriented on the March 1964 cover of Scientific American.

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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.

A185438 a(n) = 8n^2 - 2n + 1.