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A075318
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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.
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5
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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
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1,1
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COMMENTS
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(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
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LINKS
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FORMULA
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a(n) = 2*floor(n*phi^2)-1, where phi=(1+sqrt(5))/2. - Michel Dekking, Sep 08 2016
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MAPLE
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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
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MATHEMATICA
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Table[2 Floor[n ((1 + Sqrt[5]) / 2)^2] - 1, {n, 60}] (* Vincenzo Librandi, Sep 08 2016 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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