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Denominators of continued fraction convergents to sqrt(340).
2

%I #16 Jul 17 2023 00:16:27

%S 1,2,7,9,16,25,41,353,394,747,1141,1888,6805,15498,564733,1144964,

%T 3999625,5144589,9144214,14288803,23433017,201752939,225185956,

%U 426938895,652124851,1079063746,3889316089,8857695924,322766369353,654390434630

%N Denominators of continued fraction convergents to sqrt(340).

%H Vincenzo Librandi, <a href="/A041643/b041643.txt">Table of n, a(n) for n = 0..200</a>

%H <a href="/index/Rec#order_28">Index entries for linear recurrences with constant coefficients</a>, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 571538, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1).

%F G.f.: -(x^26 -2*x^25 +7*x^24 -9*x^23 +16*x^22 -25*x^21 +41*x^20 -353*x^19 +394*x^18 -747*x^17 +1141*x^16 -1888*x^15 +6805*x^14 -15498*x^13 -6805*x^12 -1888*x^11 -1141*x^10 -747*x^9 -394*x^8 -353*x^7 -41*x^6 -25*x^5 -16*x^4 -9*x^3 -7*x^2 -2*x -1)/(x^28 -571538*x^14 +1). - _Vincenzo Librandi_, Dec 22 2013

%F a(n) = 571538*a(n-14) - a(n-28) for n>27. - _Vincenzo Librandi_, Dec 22 2013

%t Denominator[Convergents[Sqrt[340], 30]] (* _Harvey P. Dale_, Jun 01 2012 *)

%t CoefficientList[Series[-(x^26 - 2 x^25 + 7 x^24 - 9 x^23 + 16 x^22 - 25 x^21 + 41 x^20 - 353 x^19 + 394 x^18 - 747 x^17 + 1141 x^16 - 1888 x^15 + 6805 x^14 - 15498 x^13 - 6805 x^12 - 1888 x^11 - 1141 x^10 - 747 x^9 - 394 x^8 - 353 x^7 - 41 x^6 - 25 x^5 - 16 x^4 - 9 x^3 - 7 x^2 - 2 x - 1)/(x^28 - 571538 x^14 + 1), {x, 0, 30}], x] (* _Vincenzo Librandi_, Dec 22 2013 *)

%o (Magma) I:=[1,2,7,9,16,25,41,353,394,747,1141,1888,6805, 15498,564733,1144964,3999625,5144589,9144214,14288803, 23433017,201752939,225185956,426938895,652124851, 1079063746,3889316089,8857695924]; [n le 28 select I[n] else 571538*Self(n-14)-Self(n-28): n in [1..40]]; // _Vincenzo Librandi_, Dec 22 2013

%Y Cf. A041642.

%K nonn,frac,easy

%O 0,2

%A _N. J. A. Sloane_.

%E More terms from _Vincenzo Librandi_, Dec 22 2013