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%I #51 Sep 01 2018 22:22:46
%S 3,4,7,11,18,119,137,256,393,649,4287,4936,9223,14159,23382,154451,
%T 177833,332284,510117,842401,5564523,6406924,11971447,18378371,
%U 30349818,200477279,230827097,431304376,662131473
%N Numerators of continued fraction convergents to sqrt(13).
%H Vincenzo Librandi, <a href="/A041018/b041018.txt">Table of n, a(n) for n = 0..200</a>
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,36,0,0,0,0,1).
%F From _Johannes W. Meijer_, Jun 12 2010: (Start)
%F a(5*n) = A006497(3*n+1),
%F a(5*n+1) = (A006497(3*n+2)-A006497(3*n+1))/2,
%F a(5*n+2) = (A006497(3*n+2)+A006497(3*n+1))/2,
%F a(5*n+3) = A006497(3*n+2),
%F a(5*n+4) = A006497(3*n+3)/2.
%F (End)
%F G.f.: (3 + 4*x + 7*x^2 + 11*x^3 + 18*x^4 + 11*x^5 - 7*x^6 + 4*x^7 - 3*x^8 + x^9)/(1 - 36*x^5 - x^10). - _Peter J. C. Moses_, Jul 29 2013
%F a(n) = A010122(n)*a(n-1)+a(n-2) with a(0)=3, a(-1)=1. - _Paul Weisenhorn_, Aug 19 2018
%p a[0]:=3: a[-1]:=1: b(0):=6: b(1):=1; b(2):=1: b(3):=1: b(4):=1:
%p for n from 1 to 100 do k:=n mod 5:
%p a[n]:=b(k)*a[n-1]+a[n-2]:
%p printf("%12d", a[n]):
%p end do: # _Paul Weisenhorn_, Aug 17 2018
%t Numerator[Convergents[Sqrt[13], 30]] (* _Vincenzo Librandi_, Oct 27 2013 *)
%t CoefficientList[Series[(3 + 4*x + 7*x^2 + 11*x^3 + 18*x^4 + 11*x^5 - 7*x^6 + 4*x^7 - 3*x^8 + x^9)/(1 - 36*x^5 - x^10),{x,0,50}],x] (* _Stefano Spezia_, Aug 31 2018 *)
%Y Cf. A010122 (continued fraction for sqrt(13)).
%Y Cf. A010470, A041019 (denominators), A041046, A041090, A041150, A041226, A041318, A041426 and A041550.
%K nonn,frac,easy
%O 0,1
%A _N. J. A. Sloane_
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