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%I #21 Sep 08 2022 08:45:07
%S 1,5,7,11,15,17,21,23,27,31,33,37,41,43,47,49,53,57,59,63,65,69,73,75,
%T 79,83,85,89,91,95,99,101,105,109,111,115,117,121,125,127,131,133,137,
%U 141,143,147,151,153,157,159,163,167,169,173,175,179,183,185,189,193
%N Pair the odd numbers such that the k-th pair is (r, r+2k) where r is the smallest odd number not included earlier: (1,3),(5,9),(7,13),(11,19),(15,25),(17,29),(21,35),(23,39),(27,45),... This is the sequence of the first member of pairs.
%C (a(n), A075318(n)) = (2A(n)-1, 2B(n)-1), where A and B are the basic Wythoff sequences A(n)=A000201(n) and B(n)=A001950(n). For a proof cf. Section 2 of the Carlitz et al. paper. - _Michel Dekking_, Sep 05 2016
%H Vincenzo Librandi, <a href="/A075317/b075317.txt">Table of n, a(n) for n = 1..1000</a>
%H L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., <a href="http://www.fq.math.ca/Scanned/10-1/carlitz1.pdf">Fibonacci representations</a>, Fib. Quart. 10 (1972), 1-28.
%F a(n) = 2*floor(n*phi)-1, where phi=(1+sqrt(5))/2. - _Michel Dekking_, Sep 05 2016
%p A075317 := proc(nmax) local r,k,a,pairs ; a := [1] ; pairs := [1,3] ; k := 2 ; r := 5 ; while nops(a) < nmax do while r in pairs do r := r+2 ; od ; a := [op(a),r] ; if r+2*k in pairs then printf("inconsistency",k) ; fi ; pairs := [op(pairs),r,r+2*k] ; k := k+1 ; od ; RETURN(a) ; end: a := A075317(200) ; for n from 1 to nops(a) do printf("%d,",op(n,a)) ; od ; # _R. J. Mathar_, Nov 12 2006
%t Table[2 Floor[n (1 + Sqrt[5]) / 2] - 1, {n, 80}] (* _Vincenzo Librandi_, Sep 05 2016 *)
%o (Magma) [2*Floor(n*(1+Sqrt(5))/2)-1: n in [1..60]]; // _Vincenzo Librandi_, Sep 05 2016
%o (Python)
%o from math import isqrt
%o def A075317(n): return (n+isqrt(5*n**2)&-2)-1 # _Chai Wah Wu_, Aug 16 2022
%Y Cf. A075318, A075319, A075320.
%K nonn
%O 1,2
%A _Amarnath Murthy_, Sep 14 2002
%E More terms from _R. J. Mathar_, Nov 12 2006
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