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%I A041008 #42 Jul 23 2021 16:58:02
%S A041008 2,3,5,8,37,45,82,127,590,717,1307,2024,9403,11427,20830,32257,149858,
%T A041008 182115,331973,514088,2388325,2902413,5290738,8193151,38063342,
%U A041008 46256493,84319835,130576328,606625147,737201475,1343826622,2081028097,9667939010,11748967107,21416906117
%N A041008 Numerators of continued fraction convergents to sqrt(7).
%H A041008 Vincenzo Librandi, <a href="/A041008/b041008.txt">Table of n, a(n) for n = 0..200</a>
%H A041008 C. Elsner, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Elsner/elsner9.html">Series of Error Terms for Rational Approximations of Irrational Numbers </a>, J. Int. Seq. 14 (2011) # 11.1.4.
%H A041008 C. Elsner, M. Stein, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Elsner2/elsner10.html">On Error Sum Functions Formed by Convergents of Real Numbers</a>, J. Int. Seq. 14 (2011) # 11.8.6.
%H A041008 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,16,0,0,0,-1).
%F A041008 G.f.: (2 + 3*x + 5*x^2 + 8*x^3 + 5*x^4 - 3*x^5 + 2*x^6 - x^7)/(1 - 16*x^4 + x^8).
%t A041008 Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[7],n]]],{n,1,50}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 16 2011 *)
%t A041008 Numerator[Convergents[Sqrt[7], 30]] (* _Vincenzo Librandi_, Oct 28 2013 *)
%t A041008 LinearRecurrence[{0,0,0,16,0,0,0,-1},{2,3,5,8,37,45,82,127},40] (* _Harvey P. Dale_, Jul 23 2021 *)
%o A041008 From _M. F. Hasler_, Nov 01 2019: (Start)
%o A041008 (PARI) A041008=contfracpnqn(c=contfrac(sqrt(7)),#c)[1,][^-1] \\ Discard possibly incorrect last element. NB: a(n)=A041008[n+1]! For more terms use:
%o A041008 A041008(n)={n<#A041008|| A041008=extend(A041008, [4, 16; 8, -1], n\.8); A041008[n+1]}
%o A041008 extend(A,c,N)={for(n=#A+1, #A=Vec(A, N), A[n]=[A[n-i]|i<-c[,1]]*c[,2]); A} \\ (End)
%Y A041008 Cf. A010465, A041009 (denominators), A266698 (quadrisection), A001081 (quadrisection).
%Y A041008 Analog for other sqrt(m): A001333 (m=2), A002531 (m=3), A001077 (m=5), A041006 (m=6), A041010 (m=8), A005667 (m=10), A041014 (m=11), A041016 (m=12), ..., A042934 (m=999), A042936 (m=1000).
%K A041008 nonn,cofr,frac,easy
%O A041008 0,1
%A A041008 _N. J. A. Sloane_

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