A075318 - OEIS

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

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OFFSET

1,1

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.

FORMULA

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

PROG

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

(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

CROSSREFS

Cf. A075317, A075319, A075320.

Sequence in context: A032415 A268044 A190745 * A171101 A287184 A197569