A075318 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 second member of pairs.

3, 9, 13, 19, 25, 29, 35, 39, 45, 51, 55, 61, 67, 71, 77, 81, 87, 93, 97, 103, 107, 113, 119, 123, 129, 135, 139, 145, 149, 155, 161, 165, 171, 177, 181, 187, 191, 197, 203, 207, 213, 217, 223, 229, 233, 239, 245, 249, 255, 259, 265, 271, 275, 281, 285, 291, 297

Offset: 1

Author

Amarnath Murthy, Sep 14 2002

Keywords

• nonn
• easy

Comments

(A075317(n),a(n)) = (2*A(n)-1, 2*B(n)-1), where A and B are the basic Wythoff sequences A(n)=A000201(n) and B(n)=A001950(n). For a proof, see section 2 of the Carlitz et al. paper. - Michel Dekking, Sep 08 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. A075317, A075319, A075320.

Programs

Magma

[2*Floor(n*((1+Sqrt(5))/2)^2)-1: n in [1..60]]; // Vincenzo Librandi, Sep 08 2016

Maple

A075318 := proc(nmax) local r,k,a,pairs ; a := [3] ; pairs := [1,3] ; k := 2 ; r := 5 ; while nops(a) < nmax do while r in pairs do r := r+2 ; od ; if r+2*k in pairs then printf("inconsistency",k) ; fi ; a := [op(a),r+2*k] ; pairs := [op(pairs),r,r+2*k] ; k := k+1 ; od ; RETURN(a) ; end: a := A075318(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)^2] - 1, {n, 60}] (* Vincenzo Librandi, Sep 08 2016 *)
2*Floor[Range[60]GoldenRatio^2]-1 (* Harvey P. Dale, Feb 08 2020 *)

PARI

a(n)=localbitprec(logint(sqrtint(45*n^4)+5*n^2,2)+2); 2*floor(n*(sqrt(5)+1)/2+n)-1 \\ Charles R Greathouse IV, Sep 09 2016

Python

from math import isqrt
def A075318(n): return (n+isqrt(5*n**2)&-2)+(n<<1)-1 # Chai Wah Wu, Aug 16 2022

Formula

a(n) = 2*floor(n*phi^2)-1, where phi=(1+sqrt(5))/2. - Michel Dekking, Sep 08 2016

Extensions

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