A293401 -id:A293401 - cp's OEIS frontend

A293402 The integer k that minimizes |k/n^2 - tau|, where tau = (1+sqrt(5))/2 (golden ratio).

0, 2, 6, 15, 26, 40, 58, 79, 104, 131, 162, 196, 233, 273, 317, 364, 414, 468, 524, 584, 647, 714, 783, 856, 932, 1011, 1094, 1180, 1269, 1361, 1456, 1555, 1657, 1762, 1870, 1982, 2097, 2215, 2336, 2461, 2589

Offset: 0

Clark Kimberling, Oct 11 2017

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

Cf. A001622, A293400, A293401.

z = 120; r = GoldenRatio;
Table[Floor[r*n^2], {n, 0, z}];   (* A293400 *)
Table[Ceiling[r*n^2], {n, 0, z}]; (* A293401 *)
Table[Round[r*n^2], {n, 0, z}];   (* A293402 *)

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

a(n) = A293400(n) if (fractional part of tau*n^2) < 1/2, else a(n) = A293401(n).

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

Original entry on oeis.org

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.

A293400 Greatest integer k such that k/n^2 < (1 + sqrt(5))/2 (the golden ratio).

Original entry on oeis.org

0, 1, 6, 14, 25, 40, 58, 79, 103, 131, 161, 195, 232, 273, 317, 364, 414, 467, 524, 584, 647, 713, 783, 855, 931, 1011, 1093, 1179, 1268, 1360, 1456, 1554, 1656, 1762, 1870, 1982, 2096, 2215, 2336, 2461, 2588

Offset: 0

Clark Kimberling, Oct 11 2017

Clark Kimberling, Table of n, a(n) for n = 0..1000
Felipe Gonçalves, Diogo Oliveira e Silva, João P. G. Ramos, New Sign Uncertainty Principles, arXiv:2003.10771 [math.CA], 2020.

Cf. A001622, A293401, A293402.

[Floor((1 + Sqrt(5))/2*n^2) : n in [0..80]]; // Wesley Ivan Hurt, Jul 03 2020

z = 120; r = GoldenRatio;
Table[Floor[r*n^2], {n, 0, z}];   (* A293400 *)
Table[Ceiling[r*n^2], {n, 0, z}]; (* A293401 *)
Table[Round[r*n^2], {n, 0, z}];   (* A293402 *)

a(n) = floor(r*n^2), where r = (1 + sqrt(5))/2.

a(n) = A293401(n) - 1 for n > 0.

A293407 Least integer k such that k/n^2 > (-1 + sqrt(5))/2 (the golden ratio).

Original entry on oeis.org

0, 1, 3, 6, 10, 16, 23, 31, 40, 51, 62, 75, 89, 105, 122, 140, 159, 179, 201, 224, 248, 273, 300, 327, 356, 387, 418, 451, 485, 520, 557, 594, 633, 674, 715, 758, 801, 847, 893, 941, 989, 1039, 1091, 1143, 1197, 1252, 1308, 1366, 1424, 1484, 1546, 1608, 1672

Offset: 0

Clark Kimberling, Oct 11 2017

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

Cf. A001622, A293401, A293403, A152738, A293408.

z = 120; r = -1+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) = ceiling(r*n^2), where r = (-1 + sqrt(5))/2.

a(n) = A152738(n) + 1 for n > 0.

cp's OEIS Frontend

A293402 The integer k that minimizes |k/n^2 - tau|, where tau = (1+sqrt(5))/2 (golden ratio).

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

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A293400 Greatest integer k such that k/n^2 < (1 + sqrt(5))/2 (the golden ratio).

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Magma

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A293407 Least integer k such that k/n^2 > (-1 + sqrt(5))/2 (the golden ratio).

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