A047452 -id:A047452 - cp's OEIS frontend

A047461 Numbers that are congruent to {1, 4} mod 8.

1, 4, 9, 12, 17, 20, 25, 28, 33, 36, 41, 44, 49, 52, 57, 60, 65, 68, 73, 76, 81, 84, 89, 92, 97, 100, 105, 108, 113, 116, 121, 124, 129, 132, 137, 140, 145, 148, 153, 156, 161, 164, 169, 172, 177, 180, 185, 188, 193, 196, 201, 204, 209, 212, 217, 220, 225, 228, 233

Offset: 1

N. J. A. Sloane

Maximal number of squares that can be covered by a queen on an n X n chessboard. - Reinhard Zumkeller, Dec 15 2008

Muniru A Asiru, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A004526, A004770, A010703, A017077, A017113, A047398, A047452.

Cf. A047470, A047524, A047535, A047615, A047617, A153125, A342819.

Filtered([1..250], n->n mod 8=1 or n mod 8 =4); # Muniru A Asiru, Jul 23 2018

[4*n-3 - ((n+1) mod 2): n in [1..70]]; // G. C. Greubel, Mar 15 2024

seq(coeff(series(factorial(n)*((8-exp(-x)+(8*x-7)*exp(x))/2), x,n+1),x,n),n=1..60); # Muniru A Asiru, Jul 23 2018

Flatten[(#+{1,4})&/@(8Range[0,30])] (* or *) LinearRecurrence[ {1,1,-1},{1,4,9},60] (* Harvey P. Dale, Jun 18 2013 *)
CoefficientList[ Series[(4x^2 + 3x + 1)/((x + 1) (x - 1)^2), {x, 0, 58}], x] (* Robert G. Wilson v, Jul 24 2018 *)

makelist(4*n -(7 + (-1)^n)/2, n, 1, 100); /* Franck Maminirina Ramaharo, Jul 22 2018 */

def A047461(n): return (n-1<<2)|(n&1) # Chai Wah Wu, Mar 30 2024

[4*n-3 - ((n+1)%2) for n in range(1,71)] # G. C. Greubel, Mar 15 2024

From R. J. Mathar, Oct 29 2008: (Start)
G.f.: x*(1+3*x+4*x^2)/((1+x)*(1-x)^2).
a(n) = a(n-2) + 8.
a(n) + a(n+1) = A004770(n).
a(n+1) - a(n) = A010703(n). (End)
a(n) = 8*floor((n-1)/2) + 4 - 3*(n mod 2). - Reinhard Zumkeller, Dec 15 2008
a(n) = A153125(n,n). - Reinhard Zumkeller, Dec 20 2008
a(n) = 8*n - a(n-1) - 11 (with a(1)=1). - Vincenzo Librandi, Aug 06 2010
a(n) = 4*n - (7 + (-1)^n)/2. - Arkadiusz Wesolowski, Sep 18 2012
a(1)=1, a(2)=4, a(3)=9, a(n) = a(n-1) + a(n-2) - a(n-3). - Harvey P. Dale, Jun 18 2013
a(n) = 1 + A004526(n)*3 + A004526(n-1)*5. - Gregory R. Bryant, Apr 16 2014
From Franck Maminirina Ramaharo, Jul 22 2018: (Start)
a(n) = A047470(n) + 1.
E.g.f.: (8 - exp(-x) + (8*x - 7)*exp(x))/2. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)+1)*Pi/16 + log(2)/4 + sqrt(2)*arccoth(sqrt(2))/8. - Amiram Eldar, Dec 11 2021

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

Original entry on oeis.org

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

A047617 Numbers that are congruent to {2, 5} mod 8.

Original entry on oeis.org

2, 5, 10, 13, 18, 21, 26, 29, 34, 37, 42, 45, 50, 53, 58, 61, 66, 69, 74, 77, 82, 85, 90, 93, 98, 101, 106, 109, 114, 117, 122, 125, 130, 133, 138, 141, 146, 149, 154, 157, 162, 165, 170, 173, 178, 181, 186, 189, 194, 197, 202, 205, 210, 213, 218, 221, 226, 229, 234

Offset: 1

N. J. A. Sloane

Numbers whose binary reflected Gray code (A014550) ends with 11. - Amiram Eldar, May 17 2021

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

Cf. A014550, A047398, A047452, A047461, A047470, A047524, A047535, A047615.

Select[Range[300],MemberQ[{2,5},Mod[#,8]]&] (* or *) LinearRecurrence[ {1,1,-1},{2,5,10},80] (* Harvey P. Dale, Feb 23 2016 *)

makelist(4*n -(5 + (-1)^n)/2, n, 1, 100); /* Franck Maminirina Ramaharo, Jul 22 2018 */

def A047617(n): return (n-1<<2)+1+(n&1) # Chai Wah Wu, Mar 30 2024

a(n) = 8*n - a(n-1) - 9 (with a(1)=2). - Vincenzo Librandi, Aug 06 2010
a(n) = 4*n - (5 + (-1)^n)/2. - Arkadiusz Wesolowski, Sep 18 2012
G.f.: (2+3*x+3*x^2)/((-1+x)^2*(1+x)). - Harvey P. Dale, Feb 23 2016
a(1)=2, a(2)=5, a(3)=10, a(n) = a(n-1) + a(n-2) - a(n-3). - Harvey P. Dale, Feb 23 2016
From Franck Maminirina Ramaharo, Jul 22 2018: (Start)
a(n) = A047470(n) + 2.
E.g.f.: (6 - exp(-x) + (8*x - 5)*exp(x))/2. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(2)*Pi/16 - log(2)/8 + sqrt(2)*log(sqrt(2)+1)/8. - Amiram Eldar, Dec 11 2021

More terms from Vincenzo Librandi, Aug 06 2010

A047524 Numbers that are congruent to {2, 7} mod 8.

Original entry on oeis.org

2, 7, 10, 15, 18, 23, 26, 31, 34, 39, 42, 47, 50, 55, 58, 63, 66, 71, 74, 79, 82, 87, 90, 95, 98, 103, 106, 111, 114, 119, 122, 127, 130, 135, 138, 143, 146, 151, 154, 159, 162, 167, 170, 175, 178, 183, 186, 191, 194, 199, 202, 207, 210, 215, 218, 223, 226, 231, 234

Offset: 1

N. J. A. Sloane

A195605 is a subsequence. - Bruno Berselli, Sep 21 2011

Muniru A Asiru, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A047398, A047461, A047452, A047470, A047535, A047615, A047617, A195605.

Filtered([0..250],n->n mod 8=2 or n mod 8=7); # Muniru A Asiru, Aug 06 2018

seq(coeff(series(x*(2+5*x+x^2)/((1+x)*(1-x)^2), x,n+1),x,n),n=1..60); # Muniru A Asiru, Aug 06 2018

Select[Range[300],MemberQ[{2,7},Mod[#,8]]&] (* or *)
LinearRecurrence[ {1,1,-1},{2,7,10},60] (* Harvey P. Dale, Nov 05 2017 *)
CoefficientList[ Series[(x^2 + 5x + 2)/((x - 1)^2 (x + 1)), {x, 0, 60}], x] (* Robert G. Wilson v, Aug 07 2018 *)

makelist(4*n - mod(n,2) - 1, n, 1, 100); /* Franck Maminirina Ramaharo, Aug 06 2018 */

is(n) = #setintersect([n%8], [2, 7]) > 0 \\ Felix Fröhlich, Aug 06 2018

def A047524(n): return (n<<2)-1-(n&1) # Chai Wah Wu, Mar 30 2024

a(n) = 8*n - a(n-1) - 7, n > 1. - Vincenzo Librandi, Aug 06 2010
From R. J. Mathar, Mar 22 2011: (Start)
a(n) = 4*n - 3/2 + (-1)^n/2.
G.f.: x*(2+5*x+x^2) / ( (1+x)*(x-1)^2 ). (End)
From Franck Maminirina Ramaharo, Aug 06 2018: (Start)
a(n) = 4*n - (n mod 2) - 1.
a(n) = A047615(n) + 2.
a(2*n) = A004771(n-1).
a(2*n-1) = A017089(n-1).
E.g.f.: ((8*x - 3)*exp(x) + exp(-x) + 2)/2. (End)
a(n) = a(n-1) + a(n-2) - a(n-3). - Muniru A Asiru, Aug 06 2018
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)+2)*Pi/16 - log(2)/8 - sqrt(2)*log(sqrt(2)+1)/8. - Amiram Eldar, Dec 11 2021

More terms from Vincenzo Librandi, Aug 06 2010

A047535 Numbers that are congruent to {4, 7} mod 8.

Original entry on oeis.org

4, 7, 12, 15, 20, 23, 28, 31, 36, 39, 44, 47, 52, 55, 60, 63, 68, 71, 76, 79, 84, 87, 92, 95, 100, 103, 108, 111, 116, 119, 124, 127, 132, 135, 140, 143, 148, 151, 156, 159, 164, 167, 172, 175, 180, 183, 188, 191, 196, 199, 204, 207, 212, 215, 220, 223, 228, 231

Offset: 1

N. J. A. Sloane

Union of A004771 and A017113.

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

Cf. A004771, A017113, A047398, A047461, A047452, A047470, A047524, A047615, A047617.

A047535:=n->4*n - (1 + (-1)^n)/2; seq(A047535(n), n=1..100); # Wesley Ivan Hurt, Feb 24 2014

Table[4n - (1 + (-1)^n)/2, {n, 100}] (* Wesley Ivan Hurt, Feb 24 2014 *)

makelist(4*n - (1 + (-1)^n)/2, n, 1, 100); /* Franck Maminirina Ramaharo, Jul 22 2018 */

def A047535(n): return (n<<2)-(n&1^1) # Chai Wah Wu, Mar 30 2024

a(n) = 8*n - a(n-1) - 5 (with a(1)=4). - Vincenzo Librandi, Aug 06 2010
a(n) = 4*n -(1 + (-1)^n)/2. - Arkadiusz Wesolowski, Sep 18 2012
G.f.: x*(4+3*x+x^2) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Jul 10 2015
From Franck Maminirina Ramaharo, Jul 22 2018: (Start)
a(n) = A047470(n) + 4.
E.g.f.: (2 - exp(-x) + (8*x - 1)*exp(x))/2. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)+1)*Pi/16 - log(2)/4 - sqrt(2)*log(sqrt(2)+1)/8. - Amiram Eldar, Dec 11 2021

A047398 Numbers that are congruent to {3, 6} mod 8.

Original entry on oeis.org

3, 6, 11, 14, 19, 22, 27, 30, 35, 38, 43, 46, 51, 54, 59, 62, 67, 70, 75, 78, 83, 86, 91, 94, 99, 102, 107, 110, 115, 118, 123, 126, 131, 134, 139, 142, 147, 150, 155, 158, 163, 166, 171, 174, 179, 182, 187, 190, 195, 198, 203, 206, 211, 214, 219, 222, 227, 230

Offset: 1

N. J. A. Sloane

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

Union of A017101 and A017137.

Cf. A047461, A047452, A047470, A047524, A047535, A047615, A047617.

A047398:=n->4*n-(3+(-1)^n)/2: seq(A047398(n), n=1..100); # Wesley Ivan Hurt, Jan 30 2017

Flatten[# + {3, 6} & /@ (8 Range[0, 28])] (* Arkadiusz Wesolowski, Sep 25 2012 *)
LinearRecurrence[{1,1,-1},{3,6,11},60] (* Harvey P. Dale, Oct 26 2020 *)

makelist(4*n + mod(n, 2) - 2, n, 1, 100); /* Franck Maminirina Ramaharo, Aug 06 2018 */

def A047398(n): return ((n<<2)|(n&1))-2 # Chai Wah Wu, Mar 30 2024

a(n) = 8*n - a(n-1) - 7, n > 1. - Vincenzo Librandi, Aug 05 2010
From R. J. Mathar, Dec 05 2011: (Start)
a(n) = 4*n - (3 + (-1)^n)/2.
G.f.: x*(3+3*x+2*x^2) / ( (1+x)*(x-1)^2 ). (End)
From Franck Maminirina Ramaharo, Aug 06 2018: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3), n > 3.
a(n) = 4*n + (n mod 2) - 2.
a(n) = A047470(n) + 3.
a(2*n) = A017137(n-1).
a(2*n-1) = A017101(n-1).
E.g.f.: ((8*x - 3)*exp(x) - exp(-x) + 4)/2. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(2)*Pi/16 + log(2)/8 - sqrt(2)*log(sqrt(2)+1)/8. - Amiram Eldar, Dec 18 2021

A047615 Numbers that are congruent to {0, 5} mod 8.

Original entry on oeis.org

0, 5, 8, 13, 16, 21, 24, 29, 32, 37, 40, 45, 48, 53, 56, 61, 64, 69, 72, 77, 80, 85, 88, 93, 96, 101, 104, 109, 112, 117, 120, 125, 128, 133, 136, 141, 144, 149, 152, 157, 160, 165, 168, 173, 176, 181, 184, 189, 192, 197, 200, 205, 208, 213, 216, 221, 224, 229, 232

Offset: 1

N. J. A. Sloane

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Union of A008590 and A004770.

Cf. A047398, A047452, A047461, A047470, A047524, A047535, A047617.

Filtered([0..250], n->n mod 8=0 or n mod 8=5); # Muniru A Asiru, Jul 23 2018

[(8*n - 7 + (-1)^n)/2 : n in [1..50]]; // Wesley Ivan Hurt, Mar 26 2015

a:=n->add(4-(-1)^j, j=1..n): seq(a(n), n=0..59); # Zerinvary Lajos, Dec 13 2008

Table[(8 n - 7 + (-1)^n)/2, {n, 1, 40}] (* Wesley Ivan Hurt, Mar 26 2015 *)
Rest@ CoefficientList[Series[x^2*(5 + 3 x)/((1 - x)^2*(1 + x)), {x, 0, 59}], x] (* Michael De Vlieger, Aug 25 2016 *)
Rest@(Range[0, 60]! CoefficientList[ Series[(6 + Exp[-x] + (8 x - 7)*Exp[x])/2, {x, 0, 60}], x]) (* or *)
LinearRecurrence[{1, 1, -1}, {0, 5, 8}, 60] (* Robert G. Wilson v, Jul 23 2018 *)

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

concat(0, Vec(x^2*(5+3*x)/((1-x)^2*(1+x)) + O(x^100))) \\ Colin Barker, Aug 25 2016

def A047615(n): return (n<<2)-3-(n&1) # Chai Wah Wu, Mar 30 2024

a(n) = 8*n-a(n-1)-11 (with a(1)=0). - Vincenzo Librandi, Aug 06 2010
a(n+1) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=5 and b(k)=2^(k+2) for k>0. - Philippe Deléham, Oct 17 2011
From Wesley Ivan Hurt, Mar 26 2015: (Start)
a(n) = a(n-1)+a(n-2)-a(n-3).
a(n) = (8n - 7 + (-1)^n)/2. (End)
G.f.: x^2*(5+3*x) / ((1-x)^2*(1+x)). - Colin Barker, Aug 25 2016
From Franck Maminirina Ramaharo, Jul 23 2018: (Start)
a(n) = A047470(n) - (-1)^(n - 1) + 1.
E.g.f.: (6 + exp(-x) + (8*x - 7)*exp(x))/2. (End)
Sum_{n>=2} (-1)^n/a(n) = log(2)/2 - (sqrt(2)-1)*Pi/16 - sqrt(2)*log(sqrt(2)+1)/8. - Amiram Eldar, Dec 18 2021

More terms from Vincenzo Librandi, Aug 06 2010

A080412 Exchange rightmost two binary digits of n > 1; a(0)=0, a(1)=2.

Original entry on oeis.org

0, 2, 1, 3, 4, 6, 5, 7, 8, 10, 9, 11, 12, 14, 13, 15, 16, 18, 17, 19, 20, 22, 21, 23, 24, 26, 25, 27, 28, 30, 29, 31, 32, 34, 33, 35, 36, 38, 37, 39, 40, 42, 41, 43, 44, 46, 45, 47, 48, 50, 49, 51, 52, 54, 53, 55, 56, 58, 57, 59, 60, 62, 61, 63, 64, 66, 65, 67, 68, 70, 69, 71, 72

Offset: 0

Reinhard Zumkeller, Feb 17 2003

Self-inverse permutation of the natural numbers: a(a(n)) = n.
Lodumo_2 of A021913. - Philippe Deléham, Apr 26 2009
The lodumo_m transformation of a list L is the list L' such that L'(n) is the smallest nonnegative integer not occurring earlier in L' and equal to L(n) (mod m). - M. F. Hasler, Dec 06 2010
From Franck Maminirina Ramaharo, Jul 20 2018: (Start)
Let
A: 0, 3,  8, 11, 16, 19, 24, 27, 32, 35, 40, 43, 48, 51, 56, 59, ... A047470
B: 1, 6,  9, 14, 17, 22, 25, 30, 33, 38, 41, 46, 49, 54, 57, 62, ... A047452
C: 2, 5, 10, 13, 18, 21, 26, 29, 34, 37, 42, 45, 50, 53, 58, 61, ... A047617
D: 4, 7, 12, 15, 20, 23, 28, 31, 36, 39, 44, 47, 52, 55, 60, 63, ... A047535.
Then the sequence is obtained by repeatedly picking terms from A,B,C,D according to the circuit A-C-B-A-D-B-C-D. The sequence begins:
A | C | B | A | D | B | C | D || A | C | B | A | D | ...
--+---+---+---+---+---+---+---++---+---+---+---+---+----
0 | 2 | 1 | 3 | 4 | 6 | 5 | 7 || 8 |10 | 9 |11 |12 | ...
(End)
The sequence is a permutation of the nonnegative integers partitioned into quadruples [4k, 4k+2, 4k+1, 4k+3] for k >= 0, i.e., the two interior terms of each quadruple are interchanged. - Guenther Schrack, Apr 22 2019

			a(20) = a('101'00') = '101'00' = 20; a(21) = a('101'01') = '101'10' = 22.
a(2) = a('10') = '01' = 1; a(3) = a('11') = '11' = 3.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).
Index entries for sequences that are permutations of the natural numbers

Cf. A004442, A007088, A021913, A080413, A080414.

Cf. A047470, A047452, A047617, A047535.

a:=[0,2,1,3,4];; for n in [6..80] do a[n]:=a[n-1]+a[n-4]-a[n-5]; od; a; # Muniru A Asiru, Jul 27 2018

R:=PowerSeriesRing(Integers(), 80); [0] cat Coefficients(R!( x*(2-x+2*x^2+x^3)/((1-x)*(1-x^4)) )); // G. C. Greubel, Apr 28 2019

A080412:=n->n+1+(1+I)*(2*I-2-(1-I)*I^(2*n)+I^(-n)-I^(1+n))/4: seq(A080412(n), n=0..100); # Wesley Ivan Hurt, May 28 2016

a[n_] := (bits = IntegerDigits[n, 2]; Join[Drop[bits, -2], {bits[[-1]], bits[[-2]]}] // FromDigits[#, 2]&); a[0]=0; a[1]=2; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Mar 11 2013 *)
ertbd[n_]:=Module[{a,b},{a,b}=TakeDrop[IntegerDigits[n,2], IntegerLength[ n,2]-2];FromDigits[Join[a,Reverse[b]],2]]; Join[{0,2},Array[ertbd,80,2]] (* The program uses the TakeDrop function from Mathematica version 10 *) (* Harvey P. Dale, Jan 07 2016 *)
CoefficientList[Series[x*(2-x+2*x^2+x^3)/((1-x)*(1-x^4)), {x,0,80}], x] (* G. C. Greubel, Apr 28 2019 *)

my(x='x+O('x^80)); concat([0], Vec(x*(2-x+2*x^2+x^3)/((1-x)*(1-x^4)))) \\ G. C. Greubel, Apr 28 2019

def A080412(n): return (0,1,-1,0)[n&3]+n # Chai Wah Wu, Jan 18 2023

(x*(2-x+2*x^2+x^3)/((1-x)*(1-x^4))).series(x, 80).coefficients(x, sparse=False) # G. C. Greubel, Apr 28 2019

a(n) = 4*floor(n/4) + a(n mod 4), for n > 3.
a(n) = a(n-1) + a(n-4) - a(n-5) for n > 4. - Joerg Arndt, Mar 11 2013
a(n) = lod_2(A021913(n)). - Philippe Deléham, Apr 26 2009
From Wesley Ivan Hurt, May 28 2016: (Start)
a(n) = n + 1 + (1+i)*(2*i-2-(1-i)*i^(2*n) + i^(-n)-i^(1+n))/4 where i=sqrt(-1).
G.f.: x*(2-x+2*x^2+x^3) / ((1-x)^2*(1+x+x^2+x^3)). (End)
E.g.f.: (sin(x) + cos(x) + (2*x + 1)*sinh(x) + (2*x - 1)*cosh(x))/2. - Ilya Gutkovskiy, May 28 2016
From Guenther Schrack, Apr 23 2019: (Start)
a(n) = (2*n - (-1)^n + (-1)^(n*(n-1)/2))/2.
a(n) = a(n-4) + 4, a(0)=0, a(1)=2, a(2)=1, a(3)=3, for n > 3. (End)

Typo in example fixed by Reinhard Zumkeller, Jul 06 2009

A075123 a(n) is the least positive integer > a(n-1) and a(n) is not 2*a(i)+a(j) for 1<=i

Original entry on oeis.org

1, 2, 3, 6, 9, 14, 17, 22, 25, 30, 33, 38, 41, 46, 49, 54, 57, 62, 65, 70, 73, 78, 81, 86, 89, 94, 97, 102, 105, 110, 113, 118, 121, 126, 129, 134, 137, 142, 145, 150, 153, 158, 161, 166, 169, 174, 177, 182, 185, 190, 193, 198, 201, 206, 209, 214, 217, 222, 225, 230

Offset: 1

Floor van Lamoen, Sep 02 2002

a(n) = A047452(n-2) for n > 3 because of first formula. - Georg Fischer, Oct 19 2018

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

Cf. A003278, A026471, A033627, A047452, A075122.

Join[{1,2,3},Table[4n-10-Mod[n,2],{n,4,60}]] (* or *)
LinearRecurrence[ {1,1,-1},{1,2,3,6,9,14},60] (* Harvey P. Dale, Oct 28 2012 *)

def A075123(n): return (n-2<<2)-2-(n&1) if n>3 else n # Chai Wah Wu, Mar 30 2024

a(n) = 4n - 10 - (n mod 2), for n>3. - Ralf Stephan, Nov 16 2004
a(n) = a(n-1) + a(n-2) - a(n-3) for n > 3. - Harvey P. Dale, Oct 28 2012
G.f.: x*(1+x+2*x^3+2*x^4+2*x^5)/((1+x)*(1-x)^2). - Georg Fischer, May 15 2019

A047401 Numbers that are congruent to {0, 1, 3, 6} mod 8.

Original entry on oeis.org

0, 1, 3, 6, 8, 9, 11, 14, 16, 17, 19, 22, 24, 25, 27, 30, 32, 33, 35, 38, 40, 41, 43, 46, 48, 49, 51, 54, 56, 57, 59, 62, 64, 65, 67, 70, 72, 73, 75, 78, 80, 81, 83, 86, 88, 89, 91, 94, 96, 97, 99, 102, 104, 105, 107, 110, 112, 113, 115, 118, 120, 121, 123

Offset: 1

N. J. A. Sloane

Partial sums of A068073. - Jeremy Gardiner, Jul 20 2013.

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

Cf. A047452, A047470, A068073.

[n : n in [0..150] | n mod 8 in [0, 1, 3, 6]]; // Wesley Ivan Hurt, Jun 01 2016

A047401:=n->2*(n-1)+(I^(n*(n-1))-1)/2: seq(A047401(n), n=1..100); # Wesley Ivan Hurt, Jun 01 2016

Select[Range[0,107], MemberQ[{0, 1, 3, 6}, Mod[#, 8]]&] (* Bruno Berselli, Dec 05 2011 *)

makelist(2*(n-1)+(%i^(n*(n-1))-1)/2,n,1,55); /* Bruno Berselli, Dec 05 2011 */

my(x='x+O('x^100)); concat(0, Vec(x^2*(1+x+2*x^2)/((x^2+1)*(x-1)^2))) \\ Altug Alkan, Jun 02 2016

G.f.: x^2*(1+x+2*x^2) / ( (x^2+1)*(x-1)^2 ). - R. J. Mathar, Dec 05 2011
a(n) = 2*(n-1)+(i^(n*(n-1))-1)/2, where i=sqrt(-1). - Bruno Berselli, Dec 05 2011
From Wesley Ivan Hurt, Jun 01 2016: (Start)
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4.
a(2k) = A047452(k), a(2k-1) = A047470(k). (End)
Sum_{n>=2} (-1)^n/a(n) = Pi/16 + (3-sqrt(2))*log(2)/8 + sqrt(2)*log(2+sqrt(2))/4. - Amiram Eldar, Dec 20 2021

cp's OEIS Frontend

A047461 Numbers that are congruent to {1, 4} mod 8.

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

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A047617 Numbers that are congruent to {2, 5} mod 8.

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A047524 Numbers that are congruent to {2, 7} mod 8.

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A047535 Numbers that are congruent to {4, 7} mod 8.

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A047398 Numbers that are congruent to {3, 6} mod 8.

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