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 1-5 of 5 results.

A181886 Lower Beatty array of the golden ratio, (1+sqrt(5))/2.

Original entry on oeis.org

0, 1, 0, 5, 3, 0, 22, 15, 4, 0, 94, 66, 21, 6, 0, 399, 282, 92, 31, 8, 0, 1691, 1197, 393, 137, 41, 9, 0, 7164, 5073, 1668, 586, 181, 47, 11, 0, 30348, 21492, 7069, 2488, 773, 208, 57, 12, 0, 128557, 91044
Offset: 1

Views

Author

Clark Kimberling, Nov 19 2010

Keywords

Comments

Upper and lower Beatty arrays are defined at A181661.
(row 1)=A049652.
(column 1)=A000004.
(column 2)=A000201 (lower Wythoff sequence).
(column 3)=u(l(n))+l(u(n)), or simply ul+lu.
(column 4)=u(ul+lu)+l(uu+ll),
whereas column 4 of the upper Beatty array is
u(uu+ll)+l(ul+lu).
U(n,k)-L(n,k)=n for n>=1, k>=0.

Examples

			Northwest corner of the array:
0....1....5....22....94....399...
0....3...15....66...282...1197...
0....4...21....92...393...1668...
0....6...31...137...586...2488...

Crossrefs

Cf. A181661, A000201, A001950, A000045.

A182787 Lower Beatty array of sqrt(3).

Original entry on oeis.org

0, 1, 0, 5, 5, 0, 23, 14, 5, 0, 103, 63, 25, 6, 0, 461, 282, 114, 31, 8, 0, 2059, 1260, 511, 142, 40, 10, 0, 9193, 5626, 2284, 638, 183, 51, 12, 0, 41039, 25116, 10199, 2853, 821, 234, 59, 13, 0, 183203, 112121, 45532, 12740, 3670, 1051
Offset: 1

Views

Author

Clark Kimberling, Dec 02 2010

Keywords

Comments

Upper and lower Beatty arrays U and L are defined in general at A181661. See also A182786. Column 2 of L=A182787 is the Beatty sequence A003511 of (1+r(3))/2, where r=sqrt(3).

Examples

			Northwest corner:
0....1....5...23...103...
0....3...14...63...282...
0....5...25..114...511...
0....6...31..142...638...

Crossrefs

Cf. A182786, A181661.

A182638 Upper Beatty array of sqrt(2).

Original entry on oeis.org

1, 3, 2, 8, 6, 3, 27, 16, 10, 4, 100, 54, 29, 13, 5, 379, 200, 102, 38, 17, 6, 1447, 759, 381, 133, 50, 20, 7, 5536, 2899, 1449, 497, 176, 59, 23, 8, 21191, 11092, 5538, 1890, 658, 208, 67, 27, 9, 81124, 42458, 21192, 7223, 2504, 779, 235, 80
Offset: 1

Views

Author

Clark Kimberling, Nov 22 2010

Keywords

Comments

Upper and lower Beatty arrays U and L are defined in general at A181661. Notation: U(n,k) and L(n,k) for n>=1 and k>=0.
Column 2 of U is A001952, the Beatty sequence of 2+sqrt(2); column 2 of L is the complement, A001951, the Beatty sequence of sqrt(2).
U(n,k)-L(n,k)=2n for all n>=1 and k>=1.

Examples

			Northwest corner of U:
1.....3.....8....27...100...
2.....6....16....54...200...
3....10....29...102...381...
4....13....38...133...497...

Crossrefs

Cf., A181661, A181886, A182639.

A182639 Lower Beatty array of sqrt(2).

Original entry on oeis.org

0, 1, 0, 6, 2, 0, 25, 12, 4, 0, 98, 50, 23, 5, 0, 377, 196, 96, 30, 7, 0, 1445, 755, 375, 125, 40, 8, 0, 5534, 2895, 1443, 489, 166, 479, 0, 21189, 11088, 5532, 1882, 648, 196, 53, 11, 0, 81122, 42454, 21186, 7215, 2494, 767, 221, 64, 12, 0, 310571
Offset: 1

Views

Author

Clark Kimberling, Nov 22 2010

Keywords

Comments

Upper and lower Beatty arrays U and L are defined in general at A181661. Notation: U(n,k) and L(n,k) for n>=1 and k>=0.

Examples

			Northwest corner of L:
0...1....6...25....98...
0...2...12...50...196...
0...4...23...96...375...
0...5...30..125...489...

Crossrefs

Cf. A181661, A181886, A182638.

A182786 Upper Beatty array of sqrt(3).

Original entry on oeis.org

1, 2, 2, 6, 4, 3, 24, 15, 7, 4, 104, 64, 27, 9, 5, 462, 283, 116, 34, 11, 6, 2060, 1261, 513, 145, 43, 14, 7, 9194, 5627, 2286, 641, 186, 55, 16, 8, 41040, 25117, 10201, 2856, 824, 238, 63, 18, 9, 183204, 112122, 45534, 12743
Offset: 1

Views

Author

Clark Kimberling, Dec 02 2010

Keywords

Comments

Upper and lower Beatty arrays U and L are defined in general at A181661. Notation: U(n,k) and L(n,k) for n>=1 and k>=0.
Column 2 of U is A022838, the Beatty sequence of sqrt(3); column 2 of L is the complement, A003511, the Beatty sequence of (1+sqrt(3))/2.
U(n,k)-L(n,k)=U(n,1)-L(n,1)=floor(n*r)-floor(n*s) where r=sqrt(3) and s=(1+r)/2.

Examples

			Northwest corner of U:
1....2....6...24...104...
2....4...15...64...283...
3....7...27..116...513...
4....9...34..145...641...

Crossrefs

Cf. A181661, A182787.
Showing 1-5 of 5 results.

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