This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A047470 #73 Feb 16 2025 08:32:39
%S A047470 0,3,8,11,16,19,24,27,32,35,40,43,48,51,56,59,64,67,72,75,80,83,88,91,
%T A047470 96,99,104,107,112,115,120,123,128,131,136,139,144,147,152,155,160,
%U A047470 163,168,171,176,179,184,187,192,195,200,203,208,211,216,219,224,227,232
%N A047470 Numbers that are congruent to {0, 3} mod 8.
%C A047470 Maximum number of squares attacked by a queen on an n X n chessboard. - _Stewart Gordon_, Mar 23 2001
%C A047470 Equivalently, maximum vertex degree in the n X n queen graph. - _Eric W. Weisstein_, Jun 20 2017
%C A047470 Number of squares attacked by a queen on a toroidal chessboard. - Diego Torres (torresvillarroel(AT)hotmail.com), May 19 2001
%C A047470 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
%C A047470 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
%H A047470 Muniru A Asiru, <a href="/A047470/b047470.txt">Table of n, a(n) for n = 1..5000</a>
%H A047470 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MaximumVertexDegree.html">Maximum Vertex Degree</a>.
%H A047470 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/QueenGraph.html">Queen Graph</a>.
%H A047470 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F A047470 a(n) = a(n-1) + 4 + (-1)^n.
%F A047470 a(n) = a(n-1) + a(n-2) - a(n-3).
%F A047470 a(n) = A042948(n) + A005843(n).
%F A047470 G.f.: (3x+5*x^2)/((1-x)*(1-x^2)).
%F A047470 a(n) = 8*n - a(n-1) - 13 (with a(1)=0). - _Vincenzo Librandi_, Aug 06 2010
%F A047470 a(n+1) = Sum_{k>=0} A030308(n,k)*A171497(k). - _Philippe Deléham_, Oct 17 2011
%F A047470 a(n) = 4*n -(9 + (-1)^n)/2. - _Arkadiusz Wesolowski_, Sep 18 2012
%F A047470 E.g.f: (10 - exp(-x) + (8*x - 9)*exp(x))/2. - _Franck Maminirina Ramaharo_, Jul 22 2018
%F A047470 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
%e A047470 From _Franck Maminirina Ramaharo_, Jul 22 2018: (Start)
%e A047470 Consider the following equivalent figure-eight knot diagrams:
%e A047470 +---------------------+ +-----------------n
%e A047470 | | | |
%e A047470 | +---------B-----+ | w-----A---e
%e A047470 | | | | | | | |
%e A047470 | n-----C---+ | | | | | |
%e A047470 | | | | | | <=> | +-------B-----s |
%e A047470 | | +---D-----+ | | | | |
%e A047470 | | | | | | | |
%e A047470 w-----A---------e | +---C-------D---------+
%e A047470 | | | |
%e A047470 s---------------------+ +-------+
%e A047470 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'.
%e A047470 Direction 'n':
%e A047470 A: 0, 3, 8, 11, 16, 19, 24, 27, 32, 35, 40, ... (this sequence);
%e A047470 B: 4, 7, 12, 15, 20, 23, 28, 31, 36, 39, 44, ... A047535;
%e A047470 C: 1, 6, 9, 14, 17, 22, 25, 30, 33, 38, 41, ... A047452;
%e A047470 D: 2, 5, 10, 13, 18, 21, 26, 29, 34, 37, 42, ... A047617.
%e A047470 Direction 's':
%e A047470 A: 0, 5, 8, 13, 16, 21, 24, 29, 32, 37, 40, ... A047615;
%e A047470 B: 1, 4, 9, 12, 17, 20, 25, 28, 33, 36, 41, ... A047461;
%e A047470 C: 2, 7, 10, 15, 18, 23, 26, 31, 34, 39, 42, ... A047524;
%e A047470 D: 3, 6, 11, 14, 19, 22, 27, 30, 35, 38, 43, ... A047398.
%e A047470 (End)
%p A047470 a:=n->add(4+(-1)^j,j=1..n):seq(a(n),n=0..64); # _Zerinvary Lajos_, Dec 13 2008
%t A047470 With[{c = 8 Range[0, 30]}, Sort[Join[c, c + 3]]] (* _Harvey P. Dale_, Oct 11 2011 *)
%t A047470 Table[(8 n - 9 - (-1)^n)/2, {n, 20}] (* _Eric W. Weisstein_, Jun 20 2017 *)
%t A047470 LinearRecurrence[{1, 1, -1}, {0, 3, 8}, 20] (* _Eric W. Weisstein_, Jun 20 2017 *)
%t A047470 CoefficientList[Series[(x (3 + 5 x))/((-1 + x)^2 (1 + x)), {x, 0, 20}], x] (* _Eric W. Weisstein_, Jun 20 2017 *)
%o A047470 (PARI) forstep(n=0,200,[3,5],print1(n", ")) \\ _Charles R Greathouse IV_, Oct 17 2011
%o A047470 (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
%o A047470 (Python)
%o A047470 def A047470(n): return (n-1<<2)-(n&1^1) # _Chai Wah Wu_, Mar 30 2024
%Y A047470 Cf. A042948, A047398, A047461, A047452, A047524, A047535, A047615, A047617.
%K A047470 nonn,easy
%O A047470 1,2
%A A047470 _N. J. A. Sloane_
%E A047470 More terms from _Vincenzo Librandi_, Aug 06 2010