Search: a293319 -id:a293319
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A293320
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Least integer k such that k/2^n > tau^2, where tau = (1+sqrt(5))/2 = golden ratio.
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+10
3
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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
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OFFSET
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0,1
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LINKS
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FORMULA
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a(n) = ceiling(r*2^n), where r = (3+sqrt(5))/2.
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MATHEMATICA
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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 *)
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PROG
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(Magma) [Ceiling((2^n*(3+Sqrt(5)))/2): n in [0..33]]; // Vincenzo Librandi, Oct 08 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A293321
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The integer k that minimizes |k/2^n - tau^2|, where tau = (1+sqrt(5))/2 = golden ratio.
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+10
3
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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
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,1
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LINKS
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FORMULA
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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).
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MATHEMATICA
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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 *)
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PROG
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(PARI) a(n) = (2^n*(3+sqrt(5))+1)\2; \\ Altug Alkan, Oct 08 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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