A171497 -id:A171497 - cp's OEIS frontend

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

N. J. A. Sloane

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

			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)

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).

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

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

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

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 *)

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

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

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

More terms from Vincenzo Librandi, Aug 06 2010

A171498 a(n) = 4*3^n-1.

Original entry on oeis.org

3, 11, 35, 107, 323, 971, 2915, 8747, 26243, 78731, 236195, 708587, 2125763, 6377291, 19131875, 57395627, 172186883, 516560651, 1549681955, 4649045867, 13947137603, 41841412811, 125524238435, 376572715307, 1129718145923, 3389154437771, 10167463313315, 30502389939947

Offset: 0

Klaus Brockhaus, Dec 10 2009

Binomial transform of A171497.

Inverse binomial transform of A171499.

Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A010703, A036543, A171497, A171499.

NestList[3#+2&,3,30]  (* Harvey P. Dale, Feb 25 2011 *)

{m=25; v=concat([3], vector(m-1)); for(n=2, m, v[n]=3*v[n-1]+2); v}

a(n) = 3*a(n-1)+2 for n > 0; a(0) = 3.
G.f.: (3-x)/((1-x)*(1-3*x)).
a(n) = A036543(n) + 2. - Philippe Deléham, Apr 13 2013
E.g.f.: exp(x)*(4*exp(2*x) - 1). - Stefano Spezia, Aug 04 2024

a(25)-a(27) from Stefano Spezia, Aug 04 2024

A010703 Period 2: repeat (3,5).

Original entry on oeis.org

3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5

Offset: 0

N. J. A. Sloane

From Klaus Brockhaus, Dec 10 2009: (Start)
Interleaving of A010701 and A010716.
Also continued fraction expansion of (15+sqrt(285))/10.
Also decimal expansion of 35/99.
Binomial transform of 3 followed by A084633 without initial terms 1,0.
Inverse binomial transform of A171497. (End)

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

Cf. A010701 (all 3's sequence), A010716 (all 5's sequence), A084633 (inverse binomial transform of repeated odd numbers), A171497.

&cat[ [3, 5]: n in [1..53] ]; // Klaus Brockhaus, Dec 10 2009

Table[If[OddQ[n], 3, 5], {n, 1, 50}] (* Stefan Steinerberger, Apr 10 2006 *)
PadRight[{},120,{3,5}] (* Harvey P. Dale, Sep 03 2012 *)

a(n)=3+n%2*2 \\ Charles R Greathouse IV, Nov 20 2011

From Klaus Brockhaus, Dec 10 2009: (Start)
a(n) = a(n-2) for n > 1; a(0) = 3, a(1) = 5.
G.f.: (3+5*x)/((1-x)*(1+x)). (End)
a(n) = 4 - (-1)^n. - Aaron J Grech, Aug 02 2024
E.g.f.: 3*cosh(x) + 5*sinh(x). - Stefano Spezia, Aug 04 2024

A171499 a(n) = 6a(n-1) - 8a(n-2) for n > 1; a(0) = 3, a(1) = 14.

Original entry on oeis.org

3, 14, 60, 248, 1008, 4064, 16320, 65408, 261888, 1048064, 4193280, 16775168, 67104768, 268427264, 1073725440, 4294934528, 17179803648, 68719345664, 274877644800, 1099511103488, 4398045462528, 17592183947264, 70368739983360

Offset: 0

Klaus Brockhaus, Dec 10 2009

Binomial transform of A171498; second binomial transform of A171497; third binomial transform of A010703.
Related to sequences A001969 and A000069, sum of each group with exponent 1. - Eric Desbiaux, Jul 24 2013
a(n) in base 2 is n+2 1s followed by n 0s. - Hussam al-Homsi, Oct 12 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (6,-8).

Cf. A010703, A171472, A171473, A171476, A171497, A171498.

[4*4^n-2^n: n in [0..30]]; // Vincenzo Librandi, Jul 18 2011

(* This program shows how A171499 arises from the Vandermonde determinant of (1,2,4,...,2^(n-1)). *)
f[j_]:= 2^j - 1; z = 15;
v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]
d[n_]:= Product[(i-1)!, {i,n}]
Table[v[n], {n,z}]                     (* A203303 *)
Table[v[n+1]/v[n], {n,z}]              (* A002884 *)
Table[v[n]*v[n+2]/(2*v[n+1])^2, {n,z}]  (* A171499 *)
(* Clark Kimberling, Jan 02 2011 *)
LinearRecurrence[{6,-8},{3,14},30] (* Harvey P. Dale, Sep 05 2021 *)

{m=23; v=concat([3, 14], vector(m-2)); for(n=3, m, v[n]=6*v[n-1]-8*v[n-2]); v}

[4^(n+1) -2^n for n in range(31)] # G. C. Greubel, Aug 31 2023

a(n) = 4*4^n - 2^n = 2^n * (2^(n+2) - 1).
G.f.: (3-4*x)/((1-2*x)*(1-4*x)).
a(n) = 4*a(n-1) + 2^n for n > 0. - Vincenzo Librandi, Jul 18 2011
a(n) = A171476(n+1)/2. - Hussam al-Homsi, Jun 06 2021
E.g.f.: 4*exp(4*x) - exp(2*x). - G. C. Greubel, Aug 31 2023

A171494 a(n) = 2*a(n-1) for n > 1; a(0) = 6, a(1) = 16.

Original entry on oeis.org

6, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648

Offset: 0

Klaus Brockhaus, Dec 10 2009

nonn

16*A000079 preceded by 6.
Binomial transform of A010726.
Inverse binomial transform of A171495.

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

Equals 2*A171497.

Cf. A000079 (powers of 2), A010726 (repeat 6, 10), A171495.

Join[{6},NestList[2#&,16,30]] (* Harvey P. Dale, Jan 13 2025 *)

{m=29; v=concat([6, 16], vector(m-2)); for(n=3, m, v[n]=2*v[n-1]); v}

a(n) = 2^(n+3) for n > 0; a(0) = 6.

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

A258935 Independence number of Keller graphs.

Original entry on oeis.org

4, 5, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592

Offset: 1

Stan Wagon, Nov 06 2015

			For G(2), a maximum independent set is {03,10,12,13,23}.

W. Jarnicki, W. Myrvold, P. Saltzman, S. Wagon, Properties, proved and conjectured, of Keller, queen, and Mycielski graphs, Ars Mathematica Contemporanea 13:2 (2017) 427-460.

Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Eric Weisstein's World of Mathematics, Independence Number
Eric Weisstein's World of Mathematics, Keller Graph

Cf. A006946, A135831, A135907.

Essentially the same as A143858, A240951, A198633, A171497, A151821, A146541 and A077552.

Mathematica
```
Join[{4, 5}, 2^Range[3, 10]]
```

a(n)=if(n>2,2^n,n+3) \\ Charles R Greathouse IV, Nov 07 2015

a(n) = 2^n except a(1) = 4 and a(2) = 5.

G.f.: x*(x*(3+2*x)-4)/(2*x-1), e.g.f.: exp(2*x)+x^2/2+2*x-1. - Benedict W. J. Irwin, Jul 15 2016

cp's OEIS Frontend

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

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A171498 a(n) = 4*3^n-1.

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A010703 Period 2: repeat (3,5).

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A171499 a(n) = 6*a(n-1) - 8*a(n-2) for n > 1; a(0) = 3, a(1) = 14.

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A171494 a(n) = 2*a(n-1) for n > 1; a(0) = 6, a(1) = 16.

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A258935 Independence number of Keller graphs.

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A171499 a(n) = 6a(n-1) - 8a(n-2) for n > 1; a(0) = 3, a(1) = 14.