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A075317 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.

%I A075317 #23 Jan 05 2025 19:51:37
%S A075317 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 A075317 79,83,85,89,91,95,99,101,105,109,111,115,117,121,125,127,131,133,137,
%U A075317 141,143,147,151,153,157,159,163,167,169,173,175,179,183,185,189,193
%N A075317 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 A075317 (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 A075317 Vincenzo Librandi, <a href="/A075317/b075317.txt">Table of n, a(n) for n = 1..1000</a>
%H A075317 L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/10-1/carlitz1.pdf">Fibonacci representations</a>, Fib. Quart. 10 (1972), 1-28.
%F A075317 a(n) = 2*floor(n*phi)-1, where phi=(1+sqrt(5))/2. - _Michel Dekking_, Sep 05 2016
%p A075317 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 A075317 Table[2 Floor[n (1 + Sqrt[5]) / 2] - 1, {n, 80}] (* _Vincenzo Librandi_, Sep 05 2016 *)
%o A075317 (Magma) [2*Floor(n*(1+Sqrt(5))/2)-1: n in [1..60]]; // _Vincenzo Librandi_, Sep 05 2016
%o A075317 (Python)
%o A075317 from math import isqrt
%o A075317 def A075317(n): return (n+isqrt(5*n**2)&-2)-1 # _Chai Wah Wu_, Aug 16 2022
%Y A075317 Cf. A075318, A075319, A075320.
%K A075317 nonn
%O A075317 1,2
%A A075317 _Amarnath Murthy_, Sep 14 2002
%E A075317 More terms from _R. J. Mathar_, Nov 12 2006