A018837 -id:A018837 - cp's OEIS frontend

A065775 Array T read by diagonals: T(i,j)=least number of knight's moves on a chessboard (infinite in all directions) needed to move from (0,0) to (i,j).

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

Offset: 0

Stewart Gordon, Dec 05 2001

			From _Clark Kimberling_, Dec 20 2010: (Start)
T(i,j) for -2<=i<=2 and -2<=j<=2:
  4 1 2 1 4=T(2,2)
  1 2 3 2 1=T(2,1)
  2 3 0 3 2=T(2,0)
  1 2 3 2 1=T(2,-1)
  4 1 2 1 4=T(2,-2)
Corner of the array, T(i,j) for i>=0, j>=0: [Corrected Oct 14 2016]
  0 3 2 3 2 3 4...
  3 2 1 2 3 4 3...
  2 1 4 3 2 3 4...
  3 2 3 2 3 4 2... (End)

Identical to A049604 except for T(1, 1).

For number of knight's moves to various subsets of the chessboard, see A018837, A183041-A183053.

From Clark Kimberling, Dec 20 2010: (Start)
T(i,j) is given in cases:
Case 1:  row 0
  T(0,0)=0, T(1,0)=3, and for m>=1,
  T(4m-2,0)=2m, T(4m-1,0)=2m+1, T(4m,0)=2m,
  T(4m+1,0)=2m+1.
Case 2:  row 1
  T(0,1)=3, T(1,1)=2, and for m>=1,
  T(4m-2,1)=2m-1, T(4m-1,1)=2m, T(4m,1)=2m+1,
  T(4m+1,1)=2m+2.
Case 3:  columns 0 and 1
  (column 0 = row 0); (column 1 = row 1).
Case 4:  For i>=2 and j>=2,
   T(i,j)=1+min{T(i-2,j-1),T(i-1,j-2)}.
Cases 1-4 determine T in the 1st quadrant;
all other T(i,j) are easily obtained by symmetry. (End)

A187180 Parse the infinite string 0101010101... into distinct phrases 0, 1, 01, 010, 10, ...; a(n) = length of n-th phrase.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Mar 06 2011

			The sequence begins
   1   1
   2   3   2   3
   4   5   4   5
   6   7   6   7
   8   9   8   9
  10  11  10  11 ...

Ray Chandler, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

See A187180-A187188 for alphabets of size 2 through 10.
See also A109337, A187199, A187200, A106249, A083219, A018837.
Essentially the same as A106249 and A018837.

1,1,seq(op(2*i*[1,1,1,1]+[0,1,0,1]), i=1..100); # Robert Israel, Oct 15 2015

Join[{1},LinearRecurrence[{1, 0, 0, 1, -1},{1, 2, 3, 2, 3},119]] (* Ray Chandler, Aug 26 2015 *)
CoefficientList[Series[(x^5 - 2 x^4 + x^3 + x^2 + 1)/((x - 1)^2 (x + 1) (x^2 + 1)), {x, 0, 150}], x] (* Vincenzo Librandi, Oct 16 2015 *)

a(n) = if(n==1, 1, (1 + (-1)^n + (1-I)*(-I)^n + (1+I)*I^n + 2*n) / 4); \\ Colin Barker, Oct 15 2015

Vec(x*(x^5-2*x^4+x^3+x^2+1) / ((x-1)^2*(x+1)*(x^2+1)) + O(x^100)) \\ Colin Barker, Oct 15 2015

Consider more generally the string 012...k012...k012...k012...k01... with an alphabet of size B, where k = B-1. The sequence begins with B 1's, and thereafter is quasi-periodic with period B^2, and increases by B in each period.
For the present example, where B=2, the sequence begins with two 1's and thereafter increases by 2 in each block of 4: (1,1) (2,3,2,3), (4,5,4,5), (6,7,6,7), ...
From Colin Barker, Oct 15 2015: (Start)
a(n) = (1+(-1)^n+(1-i)*(-i)^n+(1+i)*i^n+2*n)/4 for n>1, where i = sqrt(-1).
G.f.: x*(x^5-2*x^4+x^3+x^2+1) / ((x-1)^2*(x+1)*(x^2+1)). (End)
From Wesley Ivan Hurt, May 03 2021: (Start)
a(n) = a(n-1)+a(n-4)-a(n-5).
a(n) = floor((n+1+(-1)^floor((n+1)/2))/2) for n > 1. (End)

A183043 Triangular array, T(i,j)=number of knight's moves to points on vertical segments (n,0), (n,1),...,(n,n) on infinite chessboard.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Dec 20 2010

Stated another way, T(n,k) = distance from square (0,0) at center of an infinite open chessboard to square (n,k) via shortest knight path, for 0<=k<=n. - Fred Lunnon, May 18 2014

			Triangle starts:
0,
3,2,
2,1,4,
3,2,3,2,
2,3,2,3,4,
3,4,3,4,3,4,
4,3,4,3,4,5,4,
5,4,5,4,5,4,5,6,
4,5,4,5,4,5,6,5,6,
5,6,5,6,5,6,5,6,7,6
...
See examples under A242511.

Fred Lunnon, Knights in Daze, to appear.

Alois P. Heinz, Rows n = 0..140, flattened
Fred Lunnon, Revised tables & functions for knight's path distance and count (MAGMA code)

Cf. A065775, A183044, A183045, A183046, A018837, A018839, A242511, A242512, A242513, A242514, A242591.

// See link for recursive & explicit algorithms. - Fred Lunnon, May 18 2014

See A065775.

Edited by N. J. A. Sloane, May 23 2014

Offset corrected by Alois P. Heinz, Sep 10 2014

A083219 a(n) = n - 2*floor(n/4).

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 4, 5, 4, 5, 6, 7, 6, 7, 8, 9, 8, 9, 10, 11, 10, 11, 12, 13, 12, 13, 14, 15, 14, 15, 16, 17, 16, 17, 18, 19, 18, 19, 20, 21, 20, 21, 22, 23, 22, 23, 24, 25, 24, 25, 26, 27, 26, 27, 28, 29, 28, 29, 30, 31, 30, 31, 32, 33, 32, 33, 34, 35, 34, 35, 36, 37, 36, 37, 38

Offset: 0

Reinhard Zumkeller, Apr 22 2003

Conjecture: number of roots of P(x) = x^n - x^(n-1) - x^(n-2) - ... - x - 1 in the left half-plane. - Michel Lagneau, Apr 09 2013

a(n) is n+2 with its second least significant bit removed (see A021913(n+2) for that bit). - Kevin Ryde, Dec 13 2019

Altug Alkan, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A083220, A129756, A162751 (second highest bit removed).
Cf. A021913, A162330, A285869.
Essentially the same as A018837.

a:= n-> n - 2*floor(n/4):
seq(a(n), n=0..74);  # Alois P. Heinz, Jan 24 2021

Array[# - 2 Floor[#/4] &, 75, 0] (* Michael De Vlieger, Oct 02 2017 *)
LinearRecurrence[{1,0,0,1,-1},{0,1,2,3,2},80] (* Harvey P. Dale, Oct 01 2018 *)

a(n) = n - n\4*2 \\ Charles R Greathouse IV, Jun 11 2015

a(n) = A083220(n)/2.
a(n) = a(n-1) + n mod 2 + (n mod 4 - 1)*(1 - n mod 2), a(0) = 0.
G.f.: x*(1+x+x^2-x^3)/((1-x)^2*(1+x)*(1+x^2)). - R. J. Mathar, Aug 28 2008
a(n) = n - A129756(n). - Michel Lagneau, Apr 09 2013
Bisection: a(2*k) = 2*floor((n+2)/4), a(2*k+1) = a(2*k) + 1, k >= 0. - Wolfdieter Lang, May 08 2017
a(n) = (2*n + 3 - 2*cos(n*Pi/2) - cos(n*Pi) - 2*sin(n*Pi/2))/4. - Wesley Ivan Hurt, Oct 02 2017
a(n) = A162330(n+2) - 1 = A285869(n+3) - 1. - Kevin Ryde, Dec 13 2019
E.g.f.: ((1 + x)*cosh(x) - cos(x) + (2 + x)*sinh(x) - sin(x))/2. - Stefano Spezia, May 27 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2) - 1. - Amiram Eldar, Aug 21 2023

A106249 Expansion of (1-x+x^2+x^3)/((1+x)(1+x^2)(1-x)^2).

Original entry on oeis.org

1, 0, 1, 2, 3, 2, 3, 4, 5, 4, 5, 6, 7, 6, 7, 8, 9, 8, 9, 10, 11, 10, 11, 12, 13, 12, 13, 14, 15, 14, 15, 16, 17, 16, 17, 18, 19, 18, 19, 20, 21, 20, 21, 22, 23, 22, 23, 24, 25, 24, 25, 26, 27, 26, 27, 28, 29, 28, 29, 30, 31, 30, 31, 32, 33, 32, 33, 34, 35, 34, 35, 36, 37, 36, 37, 38

Offset: 0

Paul Barry, Apr 27 2005

Conjecture: number of roots of x^n + 1 in the left half-plane for n > 0. - Michel Lagneau, Oct 31 2012

Maximum bias of polyominoes with n+1 squares. Define the bias of a polyomino to be the difference between the number of black squares and the number of white squares when chessboard coloring is applied to the polyomino. Maximum bias for the value n is defined to be the maximum value of bias among all polyominoes of n squares. - John Mason, Dec 24 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A008611.

List([0..80],n->((n-1) mod 4)/2+(n+1)/2-1); # Muniru A Asiru, Oct 07 2018

a:= n-> n-1 - 2*floor((n-1)/4):
seq(a(n), n=0..75);  # Alois P. Heinz, Jan 24 2021

CoefficientList[Series[(1 - x + x^2 + x^3)/(1 - x - x^4 + x^5), {x, 0, 100}], x] (* Vincenzo Librandi, Dec 31 2013 *)
LinearRecurrence[{1,0,0,1,-1},{1,0,1,2,3},80] (* Harvey P. Dale, May 07 2018 *)

a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; -1,1,0,0,1]^n*[1;0;1;2;3])[1,1] \\ Charles R Greathouse IV, Sep 02 2015

G.f.: (1-x+x^2+x^3)/(1-x-x^4+x^5)=(1+x^2+2x^3+x^4+2x^5+x^6)/(1-x^4)^2.
a(n) = sum{k=0..n, -mu(k mod 4)}.
a(n) = cos(Pi*n/2)/2-sin(Pi*n/2)/2+(-1)^n/4+(2n+1)/4.
a(n) = sum{k=0..n, Jacobi(2^k, 2k+1)} [Conjecture]. - Paul Barry, Jul 23 2005
a(n) = sum{k=0..n, Product{j=1..k, ((-1)^j)^(k-j+1)}}. - Paul Barry, Nov 09 2007
a(n) = A083219(n-1). - R. J. Mathar, Aug 28 2008
a(n) = numbers of times cos(-Pi/n+2k*Pi/n) < 0 for k = 0..n-1. - Michel Lagneau, Nov 02 2012
a(n) = ((n - 1) mod 4)/2 + (n+1)/2 - 1. - John Mason, Dec 24 2013
a(n) = A018837(n-1) for n > 2. - Georg Fischer, Oct 07 2018

John Mason's contributions corrected for offset by Eric M. Schmidt, Dec 30 2013

A183041 Least number of knight's moves from (0,0) to (n,1) on infinite chessboard.

Original entry on oeis.org

3, 2, 1, 2, 3, 4, 3, 4, 5, 6, 5, 6, 7, 8, 7, 8, 9, 10, 9, 10, 11, 12, 11, 12, 13, 14, 13, 14, 15, 16, 15, 16, 17, 18, 17, 18, 19, 20, 19, 20, 21, 22, 21, 22, 23, 24, 23, 24, 25, 26, 25, 26, 27, 28, 27, 28, 29, 30, 29, 30, 31, 32, 31, 32, 33, 34, 33, 34, 35, 36

Offset: 0

Clark Kimberling, Dec 20 2010

nonn

Row 2 of the array at A065775.

Apparently a(n)=A162330(n), n>0. - R. J. Mathar, Jan 29 2011

			a(0)=3 counts (0,0) to (2,1) to (1,3) to (0,1).

Cf. A065775, A018837, A183042-A183053.

def a(n):
  if n < 2: return [3, 2][n]
  m, r = divmod(n, 4)
  return [2*m+1, 2*m+2][r%2]
print([a(n) for n in range(70)]) # Michael S. Branicky, Mar 02 2021

T(0,1)=3, T(1,1)=2, and for m>=1,
T(4m-2,1)=2m-1, T(4m-1,1)=2m, T(4m,1)=2m+1, T(4m+1,1)=2m+2.
G.f.: (2*x^5-2*x^4+x^3-x^2-x+3) / ((x-1)^2*(x+1)*(x^2+1)). - Colin Barker, Feb 19 2014

A183049 Array of least knight's moves to points (n,0), (n-1,1), ..., (1,n-1) on infinite chessboard.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Dec 22 2010

The n points (n,0), (n-1,1), ..., (1,n-1) lie in a diagonal in the first quadrant. Adjoining the matching points in the other quadrants yields the square |i|+|j|=n, as in A183051. For a description of the infinite chessboard, see A065775.

			First 6 rows (after the initial 0):
3
2 2
3 1 1
2 2 4 2
3 3 3 3 3
4 4 2 2 2 4
These numbers occupy positions on the chessboard as
indicated here, starting at the left bottom corner:
..4
..3 4
..2 3 2
..1 4 3 2
..2 1 2 3 4
0 3 2 3 2 3 4 ... (This row is A018837.)

Cf. A065775, A183050, A183051.

See A065775.

A246924 8 X 8 square array read by rows: T(i,j) = number of moves of a knight to reach the square (i,j) when starting from the square (1,2).

Original entry on oeis.org

3, 0, 3, 2, 3, 2, 3, 4, 2, 3, 2, 1, 2, 3, 4, 3, 1, 2, 1, 4, 3, 2, 3, 4, 2, 3, 2, 3, 2, 3, 4, 3, 3, 2, 3, 2, 3, 4, 3, 4, 4, 3, 4, 3, 4, 3, 4, 5, 3, 4, 3, 4, 3, 4, 5, 4, 4, 5, 4, 5, 4, 5, 4, 5

Offset: 1

Frieder Mittmann, Sep 07 2014

Cf. A018837, A246925, A065775.

A246925 8 X 8 square array read by rows: T(i,j) = number of moves of a knight to reach the square (i,j) when starting from the corner square (1,1).

Original entry on oeis.org

0, 3, 2, 3, 2, 3, 4, 5, 3, 4, 1, 2, 3, 4, 3, 4, 2, 1, 4, 3, 2, 3, 4, 5, 3, 2, 3, 2, 3, 4, 3, 4, 2, 3, 2, 3, 4, 3, 4, 5, 3, 4, 3, 4, 3, 4, 5, 4, 4, 3, 4, 3, 4, 5, 4, 5, 5, 4, 5, 4, 5, 4, 5, 6

Offset: 1

Frieder Mittmann, Sep 07 2014

The initial position needs no moves to be reached and is set to value 0
Read by rows:
a(1)   a(2) .... a(8)
a(9)  a(10) .... a(16)
   ....
   ....
a(57) a(58) .... a(64)

Cf. A018837, A246924. See A065775 for the analog on an infinite board.

cp's OEIS Frontend

A065775 Array T read by diagonals: T(i,j)=least number of knight's moves on a chessboard (infinite in all directions) needed to move from (0,0) to (i,j).

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A187180 Parse the infinite string 0101010101... into distinct phrases 0, 1, 01, 010, 10, ...; a(n) = length of n-th phrase.

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A183043 Triangular array, T(i,j)=number of knight's moves to points on vertical segments (n,0), (n,1),...,(n,n) on infinite chessboard.

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A083219 a(n) = n - 2*floor(n/4).

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A106249 Expansion of (1-x+x^2+x^3)/((1+x)*(1+x^2)*(1-x)^2).

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A183041 Least number of knight's moves from (0,0) to (n,1) on infinite chessboard.

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A183049 Array of least knight's moves to points (n,0), (n-1,1), ..., (1,n-1) on infinite chessboard.

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A246924 8 X 8 square array read by rows: T(i,j) = number of moves of a knight to reach the square (i,j) when starting from the square (1,2).

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A106249 Expansion of (1-x+x^2+x^3)/((1+x)(1+x^2)(1-x)^2).