cp's OEIS Frontend

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.

Showing 1-4 of 4 results.

A183053 Sums of knight's moves over the square |i|+|j|<=n on infinite chessboard.

Original entry on oeis.org

0, 12, 28, 48, 88, 148, 220, 312, 440, 588, 772, 1000, 1248, 1548, 1908, 2288, 2728, 3244, 3788, 4400, 5096, 5828, 6644, 7552, 8496, 9540, 10692, 11880, 13176, 14596, 16060, 17640, 19352, 21116, 23012, 25048
Offset: 0

Views

Author

Clark Kimberling, Dec 22 2010

Keywords

Comments

Partial sums matching the squares |i|+|j|=n are given by A183052.

Crossrefs

Cf. A065775, A183051, A183052.

Formula

See A065775.
Empirical g.f.: -4*x*(2*x^12-2*x^11+2*x^10-4*x^9+2*x^8-x^7-x^6-4*x^4-4*x^2-x-3) / ((x-1)^4*(x^2+1)*(x^2+x+1)^2). - Colin Barker, May 04 2014

A183050 Sums of knight's moves to points as in A183049.

Original entry on oeis.org

0, 3, 4, 5, 10, 15, 18, 23, 32, 37, 46, 57, 62, 75, 90, 95, 110, 129, 136, 153, 174, 183, 204, 227, 236, 261, 288, 297, 324, 355, 366, 395, 428, 441, 474, 509, 522, 559, 598, 611, 650, 693, 708, 749, 794, 811, 856, 903, 920, 969, 1020
Offset: 0

Views

Author

Clark Kimberling, Dec 22 2010

Keywords

Examples

			a(3)=5=3+1+1, these summands being the least numbers of knight's moves from (0,0) to the points (3,0), (2,1), (1,2) on the 3rd diagonal in the 1st quadrant - which is 1/4 of a 3rd concentric square about the origin.  See A183052 for sums over the concentric squares.

Crossrefs

Cf. A065775, A183049, A183051.

Formula

See A065775.
Empirical g.f.: x*(2*x^12-2*x^11+2*x^10-4*x^9+2*x^8-x^7-x^6-4*x^4-4*x^2-x-3) / ((x-1)^3*(x^2+1)*(x^2+x+1)^2). - Colin Barker, May 04 2014

A183052 Sums of knight's moves from (0,0) to points on the square |i|+|j|=n on infinite chessboard.

Original entry on oeis.org

0, 12, 16, 20, 40, 60, 72, 92, 128, 148, 184, 228, 248, 300, 360, 380, 440, 516, 544, 612, 696, 732, 816, 908, 944, 1044, 1152, 1188, 1296, 1420, 1464, 1580, 1712, 1764, 1896, 2036, 2088, 2236, 2392, 2444, 2600, 2772
Offset: 0

Views

Author

Clark Kimberling, Dec 22 2010

Keywords

Comments

Partial sums of A183053, which counts knight's moves from (0,0) to all points (i,j) such that |i|+|j|<=n.

Examples

			0=0
12=3+3+3+3
16=2+2+2+2+2+2+2+2
20=3+1+1+3+1+1+3+1+1+3+1+1
40=4*(3+3+3+3+3)

Crossrefs

Cf. A065775, A183050, A183051, A183053.

Formula

See A065775.
a(n) = 4*A183050(n).
Empirical g.f.: 4*x*(2*x^12-2*x^11+2*x^10-4*x^9+2*x^8-x^7-x^6-4*x^4-4*x^2-x-3) / ((x-1)^3*(x^2+1)*(x^2+x+1)^2). - Colin Barker, May 04 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

Views

Author

Clark Kimberling, Dec 22 2010

Keywords

Comments

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.

Examples

			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.)

Crossrefs

Cf. A065775, A183050, A183051.

Formula

See A065775.
Showing 1-4 of 4 results.

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