A293313
Greatest integer k such that k/2^n < (1+sqrt(5))/2 (the golden ratio).
Original entry on oeis.org
1, 3, 6, 12, 25, 51, 103, 207, 414, 828, 1656, 3313, 6627, 13254, 26509, 53019, 106039, 212078, 424157, 848315, 1696631, 3393263, 6786526, 13573052, 27146105, 54292211, 108584422, 217168845, 434337691, 868675383, 1737350766, 3474701532, 6949403065
Offset: 0
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A293313:=n->floor(2^n*(1+sqrt(5))/2): seq(A293313(n), n=0..40); # Wesley Ivan Hurt, Oct 06 2017
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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 *)
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a(n) = (2^n*(1+sqrt(5)))\2; \\ Altug Alkan, Oct 06 2017
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
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[Floor((2^n*(1+Sqrt(5))+1)/2): n in [0..33]]; // Vincenzo Librandi, Oct 08 2017
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A293315:=n->floor(1/2+2^n*(1+sqrt(5))/2): seq(A293315(n), n=0..40); # Wesley Ivan Hurt, Oct 06 2017
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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 *)
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a(n) = (2^n*(1+sqrt(5))+1)\2; \\ Altug Alkan, Oct 06 2017
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
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).
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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 *)
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a(n) = {my( m=floor(quadgen(5)<
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