cp's OEIS Frontend

A164595 a(n) = 10*a(n-1) - 17*a(n-2) for n > 1; a(0) = 5, a(1) = 37.

%I A164595 #31 Sep 08 2022 08:45:47
%S A164595 5,37,285,2221,17365,135893,1063725,8327069,65187365,510313477,
%T A164595 3994949565,31274166541,244827522805,1916614396853,15004076080845,
%U A164595 117458316061949,919513867245125,7198347299398117,56351737250814045,441145468418372461,3453475150919885845
%N A164595 a(n) = 10*a(n-1) - 17*a(n-2) for n > 1; a(0) = 5, a(1) = 37.
%C A164595 Binomial transform of A164594. Fifth binomial transform of A164737.
%H A164595 G. C. Greubel, <a href="/A164595/b164595.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..144 from Vincenzo Librandi)
%H A164595 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10,-17).
%F A164595 a(n) = ((5 + sqrt(18))*(5 + sqrt(8))^n + (5 - sqrt(18))*(5 - sqrt(8))^n)/2.
%F A164595 G.f.: (5-13*x)/(1-10*x+17*x^2).
%F A164595 E.g.f.: exp(5*x)*(5*cosh(2*sqrt(2)*x) + 3*sqrt(2)*sinh(2*sqrt(2)*x)). - _G. C. Greubel_, Aug 11 2017
%F A164595 a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*2^(n-k)*(3*Q(2*k+1) + 2*Q(2*k)), where Q(n) are the Pell-Lucas numbers (A002203). - _G. C. Greubel_, Apr 21 2020
%p A164595 seq(coeff(series( (5-13*x)/(1-10*x+17*x^2) , x, n+1), x, n), n = 0..25); # _G. C. Greubel_, Apr 21 2020
%t A164595 CoefficientList[Series[(5 -13z)/(1 -10z +17z^2), {z,0,25}], z] (* _Vladimir Joseph Stephan Orlovsky_, Jun 12 2011 *)
%t A164595 LinearRecurrence[{10,-17}, {5,37}, 25] (* _G. C. Greubel_, Aug 11 2017 *)
%o A164595 (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((5+3*r)*(5+2*r)^n+(5-3*r)*(5-2*r)^n)/2: n in [0..18] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Aug 24 2009
%o A164595 (PARI) my(x='x+O('x^25)); Vec((5-13*x)/(1-10*x+17*x^2)) \\ _G. C. Greubel_, Aug 11 2017
%o A164595 (Sage)
%o A164595 def A164595_list(prec):
%o A164595     P.<x> = PowerSeriesRing(ZZ, prec)
%o A164595     return P( (5-13*x)/(1-10*x+17*x^2) ).list()
%o A164595 A164595_list(25) # _G. C. Greubel_, Apr 21 2020
%Y A164595 Cf. A002203, A164594, A164737.
%K A164595 nonn,easy
%O A164595 0,1
%A A164595 Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009
%E A164595 Extended by _Klaus Brockhaus_ and _R. J. Mathar_, Aug 24 2009