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.

1, 5, 7, 11, 15, 17, 21, 23, 27, 31, 33, 37, 41, 43, 47, 49, 53, 57, 59, 63, 65, 69, 73, 75, 79, 83, 85, 89, 91, 95, 99, 101, 105, 109, 111, 115, 117, 121, 125, 127, 131, 133, 137, 141, 143, 147, 151, 153, 157, 159, 163, 167, 169, 173, 175, 179, 183, 185, 189, 193

Offset: 1

Author

Amarnath Murthy, Sep 14 2002

Keywords

• nonn

Comments

(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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Fibonacci representations, Fib. Quart. 10 (1972), 1-28.

Crossrefs

Cf. A075318, A075319, A075320.

Programs

Magma

[2*Floor(n*(1+Sqrt(5))/2)-1: n in [1..60]]; // Vincenzo Librandi, Sep 05 2016

Maple

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

Mathematica

Table[2 Floor[n (1 + Sqrt[5]) / 2] - 1, {n, 80}] (* Vincenzo Librandi, Sep 05 2016 *)

Python

from math import isqrt
def A075317(n): return (n+isqrt(5*n**2)&-2)-1 # Chai Wah Wu, Aug 16 2022

Formula

a(n) = 2*floor(n*phi)-1, where phi=(1+sqrt(5))/2. - Michel Dekking, Sep 05 2016

Extensions

More terms from R. J. Mathar, Nov 12 2006