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.

A183051 Array of least knight's moves to points on the square |i|+|j|=n on infinite chessboard.

Original entry on oeis.org

0, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 2, 2, 4, 2, 2, 2, 4, 2, 2, 2, 4, 2, 2, 2, 4, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 2, 2, 2, 4, 4, 4, 2, 2, 2, 4, 4, 4, 2, 2, 2, 4, 4, 4, 2, 2, 2, 4, 5, 3, 3, 3, 3, 3, 3
Offset: 1

Views

Author

Clark Kimberling, Dec 22 2010

Keywords

Examples

			Top 5 rows:
0
3 3 3 3
2 2 2 2 2 2 2 2
3 1 1 3 1 1 3 1 1 3 1 1
2 2 4 2 2 2 4 2 2 2 4 2 2 2 4 2
Row n has 4n numbers which form a square of points (i.e., unit squares) on an infinite chessboard.  The first 3 of these concentric squares are represented as follows:
....2
..2.3.2
2.3.0.3.2
..2.3.2
....2

Crossrefs

Cf. A065775, A183050, A183053.

Formula

See A065775.

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.

A183151 Number of partitions of n minus the number of primes <= n.

Original entry on oeis.org

1, 1, 1, 3, 4, 8, 11, 18, 26, 38, 51, 72, 95, 129, 170, 225, 290, 378, 482, 619, 784, 994, 1246, 1566, 1949, 2427, 3001, 3709, 4555, 5594, 6831, 8338, 10132, 12299, 14872, 17966, 21625, 26003, 31173, 37326, 44570, 53161, 63247, 75161, 89120, 105544, 124739, 147258, 173510, 204211
Offset: 1

Views

Author

Omar E. Pol, Jan 27 2011

Keywords

Comments

This has a visualization when the shell model of partition of A135010 is connected to the geometric model of the divisors of the natural numbers of A000005 which gives the location of the prime numbers for d(n)=2. See the illustrations of initial terms in the entries A000005, A000040, A000041 and A000720.

Crossrefs

Cf. A000040, A000041, A000720, A183050.

Programs

  • Mathematica
    Table[PartitionsP[n] - PrimePi[n], {n, 50}]

Formula

a(n) = A000041(n) - A000720(n) = p(n) - primepi(n).
Showing 1-4 of 4 results.

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