A293402 -id:A293402 - cp's OEIS frontend

A293401 Least integer k such that k/n^2 > (1 + sqrt(5))/2 (the golden ratio).

0, 2, 7, 15, 26, 41, 59, 80, 104, 132, 162, 196, 233, 274, 318, 365, 415, 468, 525, 585, 648, 714, 784, 856, 932, 1012, 1094, 1180, 1269, 1361, 1457, 1555, 1657, 1763, 1871, 1983, 2097, 2216, 2337, 2462, 2589

Offset: 0

Clark Kimberling, Oct 11 2017

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

Cf. A001622, A293400, A293402.

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

a(n) = A293400(n) + 1 for n > 0.

A293404 Least integer k such that k/n^2 > (3 + sqrt(5))/2 (the golden ratio).

Original entry on oeis.org

0, 3, 11, 24, 42, 66, 95, 129, 168, 213, 262, 317, 377, 443, 514, 590, 671, 757, 849, 946, 1048, 1155, 1268, 1385, 1508, 1637, 1770, 1909, 2053, 2202, 2357, 2516, 2681, 2852, 3027, 3208, 3393, 3585, 3781, 3983, 4189, 4401, 4619, 4841, 5069, 5302, 5540, 5784

Offset: 0

Clark Kimberling, Oct 11 2017

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

Cf. A001622, A293402, A293403, 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) = ceiling(r*n^2), where r = (3 + sqrt(5))/2.

a(n) = A293403(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.

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

A293401 Least integer k such that k/n^2 > (1 + sqrt(5))/2 (the golden ratio).

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A293404 Least integer k such that k/n^2 > (3 + sqrt(5))/2 (the golden ratio).

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

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

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