A047470 Numbers that are congruent to {0, 3} mod 8.

0, 3, 8, 11, 16, 19, 24, 27, 32, 35, 40, 43, 48, 51, 56, 59, 64, 67, 72, 75, 80, 83, 88, 91, 96, 99, 104, 107, 112, 115, 120, 123, 128, 131, 136, 139, 144, 147, 152, 155, 160, 163, 168, 171, 176, 179, 184, 187, 192, 195, 200, 203, 208, 211, 216, 219, 224, 227, 232

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Maximum number of squares attacked by a queen on an n X n chessboard. - Stewart Gordon, Mar 23 2001

Equivalently, maximum vertex degree in the n X n queen graph. - Eric W. Weisstein, Jun 20 2017

Number of squares attacked by a queen on a toroidal chessboard. - Diego Torres (torresvillarroel(AT)hotmail.com), May 19 2001

List of squared distances between points of diamond 'lattice' with minimal distance sqrt(3). - Arnold Neumaier (Arnold.Neumaier(AT)univie.ac.at), Aug 01 2003

Draw a figure-eight knot diagram on the plane and assign a list of nonnegative numbers at each crossing as follows. Start with 0 and choose a crossing on the knot. Pick a direction and walk around the knot, appending the following nonnegative number everytime a crossing is visited. Two series of sequences are obtained: this sequence, A047535, A047452, A047617 and A047615, A047461, A047452, A047398 (see example). - Franck Maminirina Ramaharo, Jul 22 2018

Examples

			From _Franck Maminirina Ramaharo_, Jul 22 2018: (Start)
Consider the following equivalent figure-eight knot diagrams:
+---------------------+           +-----------------n
|                     |           |                 |
|           +---------B-----+     |           w-----A---e
|           |         |     |     |           |     |   |
|     n-----C---+     |     |     |           |     |   |
|     |     |   |     |     | <=> |   +-------B-----s   |
|     |     +---D-----+     |     |   |       |         |
|     |         |           |     |   |       |         |
w-----A---------e           |     +---C-------D---------+
      |                     |         |       |
      s---------------------+         +-------+
Uppercases A,B,C,D denote crossings, and lowercases n,s,w,e denote directions. Due to symmetry and ambient isotopy, all possible sequences are obtained by starting from crossing A and choose either direction 'n' or 's'.
Direction 'n':
A: 0, 3,  8, 11, 16, 19, 24, 27, 32, 35, 40, ... (this sequence);
B: 4, 7, 12, 15, 20, 23, 28, 31, 36, 39, 44, ... A047535;
C: 1, 6,  9, 14, 17, 22, 25, 30, 33, 38, 41, ... A047452;
D: 2, 5, 10, 13, 18, 21, 26, 29, 34, 37, 42, ... A047617.
Direction 's':
A: 0, 5,  8, 13, 16, 21, 24, 29, 32, 37, 40, ... A047615;
B: 1, 4,  9, 12, 17, 20, 25, 28, 33, 36, 41, ... A047461;
C: 2, 7, 10, 15, 18, 23, 26, 31, 34, 39, 42, ... A047524;
D: 3, 6, 11, 14, 19, 22, 27, 30, 35, 38, 43, ... A047398.
(End)
		

Links

• Muniru A Asiru, Table of n, a(n) for n = 1..5000
• Eric Weisstein's World of Mathematics, Maximum Vertex Degree.
• Eric Weisstein's World of Mathematics, Queen Graph.
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Cf. A042948, A047398, A047461, A047452, A047524, A047535, A047615, A047617.

Programs

GAP

a:=[0,3,8];; for n in [4..50] do a[n]:=a[n-1]+a[n-2]-a[n-3]; od; a; # Muniru A Asiru, Jul 23 2018

Maple

a:=n->add(4+(-1)^j,j=1..n):seq(a(n),n=0..64); # Zerinvary Lajos, Dec 13 2008

Mathematica

With[{c = 8 Range[0, 30]}, Sort[Join[c, c + 3]]] (* Harvey P. Dale, Oct 11 2011 *)
Table[(8 n - 9 - (-1)^n)/2, {n, 20}] (* Eric W. Weisstein, Jun 20 2017 *)
LinearRecurrence[{1, 1, -1}, {0, 3, 8}, 20] (* Eric W. Weisstein, Jun 20 2017 *)
CoefficientList[Series[(x (3 + 5 x))/((-1 + x)^2 (1 + x)), {x, 0, 20}], x]  (* Eric W. Weisstein, Jun 20 2017 *)

PARI

forstep(n=0,200,[3,5],print1(n", ")) \\ Charles R Greathouse IV, Oct 17 2011

Python

def A047470(n): return (n-1<<2)-(n&1^1) # Chai Wah Wu, Mar 30 2024

Formula

a(n) = a(n-1) + 4 + (-1)^n.

a(n) = a(n-1) + a(n-2) - a(n-3).

a(n) = A042948(n) + A005843(n).

G.f.: (3x+5*x^2)/((1-x)*(1-x^2)).

a(n) = 8*n - a(n-1) - 13 (with a(1)=0). - Vincenzo Librandi, Aug 06 2010

a(n+1) = Sum_{k>=0} A030308(n,k)*A171497(k). - Philippe Deléham, Oct 17 2011

a(n) = 4*n -(9 + (-1)^n)/2. - Arkadiusz Wesolowski, Sep 18 2012

E.g.f: (10 - exp(-x) + (8*x - 9)*exp(x))/2. - Franck Maminirina Ramaharo, Jul 22 2018

Sum_{n>=2} (-1)^n/a(n) = (sqrt(2)-1)*Pi/16 + log(2)/2 - sqrt(2)*log(sqrt(2)+1)/8. - Amiram Eldar, Dec 11 2021

Extensions

More terms from Vincenzo Librandi, Aug 06 2010