A293404 -id:A293404 - cp's OEIS frontend

A293403 Greatest integer k such that k/n^2 < (3 + sqrt(5))/2.

0, 2, 10, 23, 41, 65, 94, 128, 167, 212, 261, 316, 376, 442, 513, 589, 670, 756, 848, 945, 1047, 1154, 1267, 1384, 1507, 1636, 1769, 1908, 2052, 2201, 2356, 2515, 2680, 2851, 3026, 3207, 3392, 3584, 3780, 3982, 4188, 4400, 4618, 4840, 5068, 5301, 5539, 5783

Offset: 0

Clark Kimberling, Oct 11 2017

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A001622, A293401, A293404, A293405.

z = 120; r = 1+GoldenRatio;
Table[Floor[r*n^2], {n, 0, z}];   (* A293403 *)
Table[Ceiling[r*n^2], {n, 0, z}]; (* A293404 *)
Table[Round[r*n^2], {n, 0, z}];   (* A293405 *)

a(n) = floor(r*n^2), where r = (3 + sqrt(5))/2.

a(n) = A293404(n) - 1 for n > 0.

A293405 The integer k that minimizes |k/n^2 - tau^2|, where tau = (1+sqrt(5))/2 (golden ratio).

Original entry on oeis.org

0, 3, 10, 24, 42, 65, 94, 128, 168, 212, 262, 317, 377, 442, 513, 589, 670, 757, 848, 945, 1047, 1155, 1267, 1385, 1508, 1636, 1770, 1909, 2053, 2202, 2356, 2516, 2681, 2851, 3026, 3207, 3393, 3584, 3780, 3982, 4189, 4401, 4618, 4841, 5069, 5302, 5540, 5783

Offset: 0

Clark Kimberling, Oct 11 2017

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A001622, A293402, A293403, A293404.

z = 120; r = 1+GoldenRatio;
Table[Floor[r*n^2], {n, 0, z}];   (* A293403 *)
Table[Ceiling[r*n^2], {n, 0, z}]; (* A293404 *)
Table[Round[r*n^2], {n, 0, z}];   (* A293405 *)

a(n) = floor(1/2 + tau*n^2).

a(n) = A293403(n) if (fractional part of (1+tau)*n^2) < 1/2, else a(n) = A293404(n).

A293408 The integer k that minimizes |k/n^2 - 1/tau|, where tau = (1+sqrt(5))/2 (golden ratio).

Original entry on oeis.org

0, 1, 2, 6, 10, 15, 22, 30, 40, 50, 62, 75, 89, 104, 121, 139, 158, 179, 200, 223, 247, 273, 299, 327, 356, 386, 418, 451, 485, 520, 556, 594, 633, 673, 714, 757, 801, 846, 892, 940, 989, 1039, 1090, 1143, 1197, 1252, 1308, 1365, 1424, 1484, 1545, 1608, 1671

Offset: 0

Clark Kimberling, Oct 11 2017

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A001622, A293402, A293404, A152738, A293407.

z = 120; r = -11+GoldenRatio;
Table[Floor[r*n^2], {n, 0, z}];   (* A152738 *)
Table[Ceiling[r*n^2], {n, 0, z}]; (* A293407 *)
Table[Round[r*n^2], {n, 0, z}];   (* A293408 *)

a(n) = floor(1/2 + (n^2)/tau).

a(n) = A152738(n) if (fractional part of (1/tau)*n^2) < 1/2, else a(n) = A293407(n).

cp's OEIS Frontend

A293403 Greatest integer k such that k/n^2 < (3 + sqrt(5))/2.

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A293405 The integer k that minimizes |k/n^2 - tau^2|, where tau = (1+sqrt(5))/2 (golden ratio).

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Formula

A293408 The integer k that minimizes |k/n^2 - 1/tau|, where tau = (1+sqrt(5))/2 (golden ratio).

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula