|
%I #10 Sep 08 2022 08:46:10
%S 1,2,4,5,6,8,9,11,12,13,15,16,17,19,20,22,23,24,26,27,29,30,31,33,34,
%T 35,37,38,40,41,42,44,45,46,48,49,51,52,53,55,56,58,59,60,62,63,64,66,
%U 67,69,70,71,73,74,76,77,78,80,81,82,84,85,87,88,89,91
%N Floor(r*n), where r = (5 - sqrt(5))/2; the Beatty complement of A003231.
%C Let r = (5 - sqrt(5))/2 and s = (5 + sqrt(5))/2. Then 1/r + 1/s = 1, so that A249115 and A003231 are a pair of complementary Beatty sequences. Let tau = (1 + sqrt(5))/2, the golden ratio. Let R = {h*tau, h >= 1} and S = {k*(tau - 1), k >= 1}. Then A249115(n) is the position of n*(tau - 1) in the ordered union of R and S.
%H Clark Kimberling, <a href="/A249115/b249115.txt">Table of n, a(n) for n = 1..10000</a>
%H Scott V. Tezlaf, <a href="https://arxiv.org/abs/1806.00331">On ordinal dynamics and the multiplicity of transfinite cardinality</a>, arXiv:1806.00331 [math.NT], 2018. See p. 9.
%t Table[Floor[(5 - Sqrt[5])/2*n], {n, 1, 200}]
%o (Magma) [Floor(n*(5-Sqrt(5))/2): n in [1..100]]; // _Vincenzo Librandi_, Oct 25 2014
%Y Cf. A003231, A001622.
%K nonn,easy
%O 1,2
%A _Clark Kimberling_, Oct 21 2014
|
