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A002965
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Interleave denominators (A000129) and numerators (A001333) of convergents to sqrt(2).
(Formerly M0671)
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29
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0, 1, 1, 1, 2, 3, 5, 7, 12, 17, 29, 41, 70, 99, 169, 239, 408, 577, 985, 1393, 2378, 3363, 5741, 8119, 13860, 19601, 33461, 47321, 80782, 114243, 195025, 275807, 470832, 665857, 1136689, 1607521, 2744210, 3880899, 6625109, 9369319, 15994428, 22619537
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OFFSET
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0,5
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COMMENTS
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Denominators of Farey fraction approximations to sqrt(2). The fractions are 1/0, 0/1, 1/1, 2/1, 3/2, 4/3, 7/5, 10/7, 17/12, .... See A082766(n+2) or A119016 for the numerators. "Add" (meaning here to add the numerators and add the denominators, not to add the fractions) 1/0 to 1/1 to make the fraction bigger: 2/1. Now 2/1 is too big, so add 1/1 to make the fraction smaller: 3/2, 4/3. Now 4/3 is too small, so add 3/2 to make the fraction bigger: 7/5, 10/7, ... Because the continued fraction for sqrt(2) is all 2's, it will always take exactly two terms here to switch from a number that's bigger than sqrt(2) to one that's less. A097545/A097546 gives the similar sequence for Pi. A119014/A119015 gives the similar sequence for e. - Joshua Zucker, May 09 2006
The principal and intermediate convergents to 2^(1/2) begin with 1/1, 3/2 4/3, 7/5, 10/7; essentially, numerators=A143607, denominators=A002965. - Clark Kimberling, Aug 27 2008
((a(2n)*a(2n+1))^2 is a triangular square. - Hugh Darwen, Feb 23 2012
a(2n) are the interleaved values of m such that 2*m^2+1 and 2*m^2-1 are squares, respectively; a(2n+1) are the interleaved values of their corresponding integer square roots. - Richard R. Forberg, Aug 19 2013
Coefficients of (sqrt(2)+1)^n are a(2n)*sqrt(2)+a(2n+1). - John Molokach, Nov 29 2015
Apart from the first two terms, this is the sequence of denominators of the convergents of the continued fraction expansion sqrt(2) = 1/(1 - 1/(2 + 1/(1 - 1/(2 + 1/(1 - ....))))). - Peter Bala, Feb 02 2017
Limit_{n->infinity} a(2n+1)/a(2n) = sqrt(2); lim_{n->infinity} a(2n)/a(2n-1) = (2+sqrt(2))/2. - Ctibor O. Zizka, Oct 28 2018
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REFERENCES
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C. Brezinski, History of Continued Fractions and Padé Approximants. Springer-Verlag, Berlin, 1991, p. 24.
Jay Kappraff, Musical Proportions at the Basis of Systems of Architectural Proportion both Ancient and Modern, in Volume I of K. Williams and M.J. Ostwald (eds.), Architecture and Mathematics from Antiquity to the Future, DOI 10.1007/978-3-319-00143-2_27, Springer International Publishing Switzerland 2015. See Eq. 32.7.
Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Guelena Strehler, Chess Fractal, April 2016, p. 24.
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LINKS
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FORMULA
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a(n) = 2*a(n-2) + a(n-4) if n>3; a(0)=0, a(1)=a(2)=a(3)=1.
a(2*n) = a(2*n-1) + a(2*n-2) and a(2*n+1) = 2*a(2*n) - a(2*n-1).
G.f.: (x+x^2-x^3)/(1-2*x^2-x^4).
For n > 0, a(2*n) = a(2*n-1) + a(2*n-2) and a(2*n+1) = a(2*n) + a(2*n-2). - Jon Perry, Sep 12 2012
a(n) = (((sqrt(2) - 2)*(-1)^n + 2 + sqrt(2))*(1 + sqrt(2))^(floor(n/2)) - ((2 + sqrt(2))*(-1)^n -2 + sqrt(2))*(1 - sqrt(2))^(floor(n/2)))/8. - Ilya Gutkovskiy, Jul 18 2016
a(n) = a(n-1) + a(n-2-(n mod 2)); a(0)=0, a(1)=1. - Ctibor O. Zizka, Oct 28 2018
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EXAMPLE
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The convergents are rational numbers given by the recurrence relation p/q -> (p + 2*q)/(p + q). Starting with 1/1, the next three convergents are (1 + 2*1)/(1 + 1) = 3/2, (3 + 2*2)/(3 + 2) = 7/5, and (7 + 2*5)/(7 + 5) = 17/12. The sequence puts the denominator first, so a(2) through a(9) are 1, 1, 2, 3, 5, 7, 12, 17. - Michael B. Porter, Jul 18 2016
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MAPLE
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A002965 := proc(n) option remember; if n <= 0 then 0; elif n <= 3 then 1; else 2*A002965(n-2)+A002965(n-4); fi; end;
A002965:=-(1+2*z+z**2+z**3)/(-1+2*z**2+z**4); # conjectured by Simon Plouffe in his 1992 dissertation; gives sequence except for two leading terms
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MATHEMATICA
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With[{c=Convergents[Sqrt[2], 20]}, Join[{0, 1}, Riffle[Denominator[c], Numerator[c]]]] (* Harvey P. Dale, Oct 03 2012 *)
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PROG
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(PARI) a(n)=if(n<4, n>0, 2*a(n-2)+a(n-4))
(PARI) x='x+O('x^100); concat(0, Vec((x+x^2-x^3)/(1-2*x^2-x^4))) \\ Altug Alkan, Dec 04 2015
(JavaScript)
a=new Array(); a[0]=0; a[1]=1;
for (i=2; i<50; i+=2) {a[i]=a[i-1]+a[i-2]; a[i+1]=a[i]+a[i-2]; }
(Haskell)
import Data.List (transpose)
a002965 n = a002965_list !! n
a002965_list = concat $ transpose [a000129_list, a001333_list]
(Magma) I:=[0, 1, 1, 1]; [n le 4 select I[n] else 2*Self(n-2)+Self(n-4): n in [1..50]]; // Vincenzo Librandi, Nov 30 2015
(GAP) a:=[0, 1];; for n in [3..45] do a[n]:=a[n-1]+a[n-2-((n-1) mod 2)]; od; a; # Muniru A Asiru, Oct 28 2018
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CROSSREFS
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KEYWORD
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nonn,easy,nice,frac
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AUTHOR
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EXTENSIONS
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Thanks to Michael Somos for several comments which improved this entry.
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STATUS
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approved
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