A293313 -id:A293313 - cp's OEIS frontend

A293314 Least integer k such that k/2^n > (1+sqrt(5))/2 (the golden ratio).

2, 4, 7, 13, 26, 52, 104, 208, 415, 829, 1657, 3314, 6628, 13255, 26510, 53020, 106040, 212079, 424158, 848316, 1696632, 3393264, 6786527, 13573053, 27146106, 54292212, 108584423, 217168846, 434337692, 868675384, 1737350767, 3474701533, 6949403066

Offset: 0

Clark Kimberling, Oct 06 2017

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

Cf. A001622, A293313, A293315.

A293314:=n->ceil(2^n*(1+sqrt(5))/2): seq(A293314(n), n=0..40); # Wesley Ivan Hurt, Oct 06 2017

z = 120; r = GoldenRatio;
Table[Floor[r*2^n], {n, 0, z}];   (* A293313 *)
Table[Ceiling[r*2^n], {n, 0, z}]; (* A293314 *)
Table[Round[r*2^n], {n, 0, z}];   (* A293315 *)

a(n) = ceil(2^n*(1+sqrt(5))/2) \\ Altug Alkan, Oct 06 2017

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

a(n) = A293313(n) + 1.

A293315 The integer k that minimizes |k/2^n - r|, where r = golden ratio.

Original entry on oeis.org

2, 3, 6, 13, 26, 52, 104, 207, 414, 828, 1657, 3314, 6627, 13255, 26510, 53020, 106039, 212079, 424158, 848316, 1696632, 3393263, 6786526, 13573053, 27146106, 54292211, 108584423, 217168846, 434337692, 868675383, 1737350766, 3474701533, 6949403065

Offset: 0

Clark Kimberling, Oct 06 2017

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

Cf. A001622, A293313, A293314.

[Floor((2^n*(1+Sqrt(5))+1)/2): n in [0..33]]; // Vincenzo Librandi, Oct 08 2017

A293315:=n->floor(1/2+2^n*(1+sqrt(5))/2): seq(A293315(n), n=0..40); # Wesley Ivan Hurt, Oct 06 2017

z = 120; r = GoldenRatio;
Table[Floor[r*2^n], {n, 0, z}];   (* A293313 *)
Table[Ceiling[r*2^n], {n, 0, z}]; (* A293314 *)
Table[Round[r*2^n], {n, 0, z}];   (* A293315 *)

a(n) = (2^n*(1+sqrt(5))+1)\2; \\ Altug Alkan, Oct 06 2017

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

a(n) = A293313(n) if (fractional part of r*2^n) < 1/2, else a(n) = A293313(n).

A293319 Greatest integer k such that k/2^n < tau^2, where tau = (1+sqrt(5))/2 = golden ratio.

Original entry on oeis.org

2, 5, 10, 20, 41, 83, 167, 335, 670, 1340, 2680, 5361, 10723, 21446, 42893, 85787, 171575, 343150, 686301, 1372603, 2745207, 5490415, 10980830, 21961660, 43923321, 87846643, 175693286, 351386573, 702773147, 1405546295, 2811092590, 5622185180, 11244370361

Offset: 0

Clark Kimberling, Oct 07 2017

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

Cf. A001622, A293313, A293320, A293321.

[Floor((2^n*(3+Sqrt(5)))/2): n in [0..33]]; // Vincenzo Librandi, Oct 08 2017

z = 120; r = 1+GoldenRatio;
Table[Floor[r*2^n], {n, 0, z}];   (* A293319 *)
Table[Ceiling[r*2^n], {n, 0, z}]; (* A293320 *)
Table[Round[r*2^n], {n, 0, z}];   (* A293321 *)

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

a(n) = A293320(n) - 1.

A293320 Least integer k such that k/2^n > tau^2, where tau = (1+sqrt(5))/2 = golden ratio.

Original entry on oeis.org

3, 6, 11, 21, 42, 84, 168, 336, 671, 1341, 2681, 5362, 10724, 21447, 42894, 85788, 171576, 343151, 686302, 1372604, 2745208, 5490416, 10980831, 21961661, 43923322, 87846644, 175693287, 351386574, 702773148, 1405546296, 2811092591, 5622185181, 11244370362

Offset: 0

Clark Kimberling, Oct 07 2017

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

Cf. A001622, A293313, A293319, A293321.

[Ceiling((2^n*(3+Sqrt(5)))/2): n in [0..33]]; // Vincenzo Librandi, Oct 08 2017

z = 120; r = 1+GoldenRatio;
Table[Floor[r*2^n], {n, 0, z}];   (* A293319 *)
Table[Ceiling[r*2^n], {n, 0, z}]; (* A293320 *)
Table[Round[r*2^n], {n, 0, z}];   (* A293321 *)

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

a(n) = A293319(n) + 1.

A293321 The integer k that minimizes |k/2^n - tau^2|, where tau = (1+sqrt(5))/2 = golden ratio.

Original entry on oeis.org

3, 5, 10, 21, 42, 84, 168, 335, 670, 1340, 2681, 5362, 10723, 21447, 42894, 85788, 171575, 343151, 686302, 1372604, 2745208, 5490415, 10980830, 21961661, 43923322, 87846643, 175693287, 351386574, 702773148, 1405546295, 2811092590, 5622185181, 11244370361

Offset: 0

Clark Kimberling, Oct 07 2017

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

Cf. A001622, A293313, A293319, A293320.

z = 120; r = 1+GoldenRatio;
Table[Floor[r*2^n], {n, 0, z}];   (* A293319 *)
Table[Ceiling[r*2^n], {n, 0, z}]; (* A293320 *)
Table[Round[r*2^n], {n, 0, z}]; (* A293321 *)

a(n) = (2^n*(3+sqrt(5))+1)\2; \\ Altug Alkan, Oct 08 2017

a(n) = floor(1/2 + r*2^n), where r = (3+sqrt(5))/2.

a(n) = A293319(n) if (fractional part of r*2^n) < 1/2, else a(n) = A293316(n).

A293322 Greatest integer k such that k/2^n < 1/tau, where tau = (1+sqrt(5))/2 = golden ratio.

Original entry on oeis.org

0, 1, 2, 4, 9, 19, 39, 79, 158, 316, 632, 1265, 2531, 5062, 10125, 20251, 40503, 81006, 162013, 324027, 648055, 1296111, 2592222, 5184444, 10368889, 20737779, 41475558, 82951117, 165902235, 331804471, 663608942, 1327217884, 2654435769, 5308871538

Offset: 0

Clark Kimberling, Oct 07 2017

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

Cf. A001622, A293313, A293323, A293324.

z = 120; r = -1+GoldenRatio;
Table[Floor[r*2^n], {n, 0, z}];   (* A293322 *)
Table[Ceiling[r*2^n], {n, 0, z}]; (* A293323 *)
Table[Round[r*2^n], {n, 0, z}]; (* A293324 *)

a(n) = 2^n*(sqrt(5)-1)\2; \\ Altug Alkan, Oct 08 2017

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

a(n) = A293323(n) - 1.

A293323 Least integer k such that k/2^n > 1/tau, where tau = (1+sqrt(5))/2 = golden ratio.

Original entry on oeis.org

1, 2, 3, 5, 10, 20, 40, 80, 159, 317, 633, 1266, 2532, 5063, 10126, 20252, 40504, 81007, 162014, 324028, 648056, 1296112, 2592223, 5184445, 10368890, 20737780, 41475559, 82951118, 165902236, 331804472, 663608943, 1327217885, 2654435770, 5308871539

Offset: 0

Clark Kimberling, Oct 07 2017

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

Cf. A001622, A293313, A293322, A293324.

z = 120; r = -1+GoldenRatio;
Table[Floor[r*2^n], {n, 0, z}];   (* A293322 *)
Table[Ceiling[r*2^n], {n, 0, z}]; (* A293323 *)
Table[Round[r*2^n], {n, 0, z}];   (* A293324 *)

a(n) = ceil(2^(n-1)*(sqrt(5)-1)); \\ Altug Alkan, Oct 08 2017

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

a(n) = A293322(n) + 1.

A376448 a(n) = k if k is odd otherwise a(n) = k+1 and k = floor( 2^n*(1+sqrt(5))/2 ).

Original entry on oeis.org

1, 3, 7, 13, 25, 51, 103, 207, 415, 829, 1657, 3313, 6627, 13255, 26509, 53019, 106039, 212079, 424157, 848315, 1696631, 3393263, 6786527, 13573053, 27146105, 54292211, 108584423, 217168845, 434337691, 868675383, 1737350767, 3474701533, 6949403065, 13898806131, 27797612261, 55595224523

Offset: 0

Thomas Scheuerle, Sep 23 2024

The sequence of all multiples of an irrational b is equidistributed modulo 1. Such a sequence is called a Weyl sequence. It is common practice in computing to approximate a Weyl sequence by taking integer multiples of some integer m modulo a power of two. This requires that the integer m is odd. This sequence provides suitable m = a(n) for the case modulo 2^n. It utilizes the golden ratio for approximation of irrationality.

			An example for a pseudo Weyl sequence obtained from a(3):
{0, 1, 2, 3, 4, 5, 6, 7} * a(3) mod 2^3 = {0, 5, 2, 7, 4, 1, 6, 3}. (Without zero also part of A194868).

Cf. A054065, A194868, A293313, A293314, A293315.

k[n_]:=Floor[2^n*GoldenRatio];Table[If[OddQ[k[n]],k[n],k[n]+1],{n,0,35}] (* James C. McMahon, Oct 20 2024 *)

PARI
```
a(n) = {my( m=floor(quadgen(5)<
				
```

a(n) = 2*A293313(n-1) + 1, for n > 0.

cp's OEIS Frontend

A293314 Least integer k such that k/2^n > (1+sqrt(5))/2 (the golden ratio).

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A293315 The integer k that minimizes |k/2^n - r|, where r = golden ratio.

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A293319 Greatest integer k such that k/2^n < tau^2, where tau = (1+sqrt(5))/2 = golden ratio.

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A293320 Least integer k such that k/2^n > tau^2, where tau = (1+sqrt(5))/2 = golden ratio.

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A293321 The integer k that minimizes |k/2^n - tau^2|, where tau = (1+sqrt(5))/2 = golden ratio.

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A293322 Greatest integer k such that k/2^n < 1/tau, where tau = (1+sqrt(5))/2 = golden ratio.

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A293323 Least integer k such that k/2^n > 1/tau, where tau = (1+sqrt(5))/2 = golden ratio.

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A376448 a(n) = k if k is odd otherwise a(n) = k+1 and k = floor( 2^n*(1+sqrt(5))/2 ).

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