A075317 - OEIS

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

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OFFSET

1,2

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.

FORMULA

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

PROG

(Magma) [2*Floor(n*(1+Sqrt(5))/2)-1: n in [1..60]]; // 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

CROSSREFS

Cf. A075318, A075319, A075320.

Sequence in context: A037141 A355013 A029624 * A134643 A039001 A023741

Adjacent sequences: A075314 A075315 A075316 * A075318 A075319 A075320

KEYWORD