A002965 Interleave denominators (A000129) and numerators (A001333) of convergents to sqrt(2).

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice
• frac

Comments

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

Examples

			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
		

References

• 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.

Links

• T. D. Noe, Table of n, a(n) for n = 0..500
• Damanvir Singh Binner, Proofs of Chappelon and Alfonsín Conjectures On Square Frobenius Numbers and its Relationship to Simultaneous Pell's Equations, arXiv:2112.15474 [math.NT], 2021.
• Jonathan Chappelon and Jorge Luis Ramírez Alfonsín, The Square Frobenius Number, arXiv:2006.14219 [math.NT], 2020.
• H. S. M. Coxeter, The role of intermediate convergents in Tait's explanation for phyllotaxis, J. Algebra 20 (1972), 167-175.
• Clark Kimberling, Best lower and upper approximates to irrational numbers, Elemente der Mathematik, 52 (1997) 122-126.
• Pierre Lamothe, En marge du problème des cercles tangents
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• Dave Rusin, Farey fractions on sci.math [Broken link]
• Dave Rusin, Farey fractions on sci.math [Cached copy]
• K. Williams, The sacred cult revisited: the pavement of the baptistery of San Giovanni, Florence, Math. Intellig., 16 (No. 2, 1994), 18-24.
• Index entries for linear recurrences with constant coefficients, signature (0,2,0,1).

Crossrefs

Cf. A000129(n) = a(2n), A001333(n) = a(2n+1).

Cf. A001109, A155046.

Programs

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

Haskell

import Data.List (transpose)
a002965 n = a002965_list !! n
a002965_list = concat $ transpose [a000129_list, a001333_list]
-- Reinhard Zumkeller, Jan 01 2014

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];}
document.write(a); // Jon Perry, Sep 12 2012

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

Maple

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

Mathematica

LinearRecurrence[{0, 2, 0, 1}, {0, 1, 1, 1}, 42] (* Vladimir Joseph Stephan Orlovsky, Feb 13 2012 *)
With[{c=Convergents[Sqrt[2],20]},Join[{0,1},Riffle[Denominator[c], Numerator[c]]]] (* Harvey P. Dale, Oct 03 2012 *)

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

Formula

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).

a(0)=0, a(1)=1, a(n) = a(n-1) + a(2*[(n-2)/2]). - Franklin T. Adams-Watters, Jan 31 2006

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

Extensions

Thanks to Michael Somos for several comments which improved this entry.