A041463 - OEIS

%I #17 Nov 29 2022 16:20:42

%S 1,1,3,4,7,67,74,733,807,1540,3887,5427,166697,172124,510945,683069,

%T 1194014,11429195,12623209,125038076,137661285,262699361,663060007,

%U 925759368,28435841047,29361600415,87159041877

%N Denominators of continued fraction convergents to sqrt(247).

%H Vincenzo Librandi, <a href="/A041463/b041463.txt">Table of n, a(n) for n = 0..200</a>

%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,0,0,0,170584,0,0,0,0,0,0,0,0,0,0,0,-1).

%F G.f.: -(x^22 -x^21 +3*x^20 -4*x^19 +7*x^18 -67*x^17 +74*x^16 -733*x^15 +807*x^14 -1540*x^13 +3887*x^12 -5427*x^11 -3887*x^10 -1540*x^9 -807*x^8 -733*x^7 -74*x^6 -67*x^5 -7*x^4 -4*x^3 -3*x^2 -x -1)/(x^24 -170584*x^12 +1). - _Vincenzo Librandi_, Dec 18 2013

%F a(n) = 170584*a(n-12) - a(n-24) for n>23. - _Vincenzo Librandi_, Dec 18 2013

%t Denominator[Convergents[Sqrt[247], 30]] (* or *) CoefficientList[Series[-(x^22 - x^21 + 3 x^20 - 4 x^19 + 7 x^18 - 67 x^17 + 74 x^16 - 733 x^15 + 807 x^14 - 1540 x^13 + 3887 x^12 - 5427 x^11 - 3887 x^10 - 1540 x^9 - 807 x^8 - 733 x^7 - 74 x^6 - 67 x^5 - 7 x^4 - 4 x^3 - 3 x^2 - x - 1)/(x^24 - 170584 x^12 + 1), {x, 0, 30}], x] (* _Vincenzo Librandi_, Dec 18 2013 *)

%t LinearRecurrence[{0,0,0,0,0,0,0,0,0,0,0,170584,0,0,0,0,0,0,0,0,0,0,0,-1},{1,1,3,4,7,67,74,733,807,1540,3887,5427,166697,172124,510945,683069,1194014,11429195,12623209,125038076,137661285,262699361,663060007,925759368},30] (* _Harvey P. Dale_, Nov 29 2022 *)

%o (Magma) I:=[1,1,3,4,7,67,74,733,807,1540,3887,5427, 166697,172124,510945,683069,1194014,11429195,12623209, 125038076,137661285,262699361,663060007,925759368]; [n le 24 select I[n] else 170584*Self(n-12)-Self(n-24): n in [1..40]]; // _Vincenzo Librandi_, Dec 18 2013

%Y Cf. A041462.

%K nonn,frac,easy

%O 0,3

%A _N. J. A. Sloane_.