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 A153316 Numerators of continued fraction convergents to sqrt(5/4). 0
 1, 9, 19, 161, 341, 2889, 6119, 51841, 109801, 930249, 1970299, 16692641, 35355581, 299537289, 634430159, 5374978561, 11384387281, 96450076809, 204284540899, 1730726404001, 3665737348901 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS In general, denominators, a(k,n) and numerators, b(k,n), of continued fraction convergents to sqrt((k+1)/k) may be found as follows: a(k,0) = 1, a(k,1) = 2k; for n>0, a(k,2n)=2*a(k,2n-1)+a(k,2n-2) and a(k,2n+1)=(2k)*a(k,2n)+a(k,2n-1); b(k,0) = 1, b(k,1) = 2k+1; for n>0, b(k,2n)=2*b(k,2n-1)+b(k,2n-2) and b(k,2n+1)=(2k)*b(k,2n)+b(k,2n-1). For example, the convergents to sqrt(4/3) start 1/1, 9/8, 19/17, 161/144, 341/305. In general, if a(k,n) and b(k,n) are the denominators and numerators, respectively, of continued fraction convergents to sqrt((k+1)/k) as defined above, then k*a(k,2n)^2-a(k,2n-1)*a(k,2n+1)=k=k*a(k,2n-2)*a(k,2n)-a(k,2n-1)^2 and b(k,2n-1)*b(k,2n+1)-k*b(k,2n)^2=k+1=b(k,2n-1)^2-k*b(k,2n-2)*b(k,2n); for example, if k=4 and n=3, then b(4,n)=a(n) and 4*a(4,6)^2-a(4,5)*a(4,7)=4*5473^2-2584*46368=4; 4*a(4,4)*a(4,6)-a(4,5)^2=4*305*5473-2584^2=4; b(4,5)*b(4,7)-4*b(4,6)^2=2889*51841-4*6119^2=5; b(4,5)^2-4*b(4,4)*b(4,6)=2889^2-4*341*6119=5. LINKS FORMULA For n>0, a(2n)=2a(2n-1)+a(2n-2) and a(2n+1)=8a(2n)+a(2n-1). Empirical G.f.: (1+9*x+x^2-x^3)/(1-18*x^2+x^4). [Colin Barker, Jan 01 2012] EXAMPLE The initial convergents are 1, 9/8, 19/17, 161/144, 341/305, 2889/2584, 6119/5473, 51841/46368, 109801/98209, 930249/832040, 1970299/1762289, CROSSREFS Cf. A000129, A001333, A142238-A142239, A153313-153318. Sequence in context: A177130 A177179 A041677 * A041160 A089565 A001154 Adjacent sequences:  A153313 A153314 A153315 * A153317 A153318 A153319 KEYWORD nonn AUTHOR Charlie Marion, Jan 07 2009 STATUS approved

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