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%I #21 Sep 08 2022 08:45:47
%S 1,11,93,743,5849,45859,359157,2811967,22014001,172336571,1349127693,
%T 10561555223,82680381449,647257375699,5067007272357,39666697336687,
%U 310527849736801,2430944642644331,19030472980917693,148978670884223303
%N a(n) = 10*a(n-1) - 17*a(n-2) for n > 1; a(0) = 1, a(1) = 11.
%C Binomial transform of A164546.
%C Fifth binomial transform of A164640.
%H Vincenzo Librandi, <a href="/A164547/b164547.txt">Table of n, a(n) for n = 0..144</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10,-17).
%F a(n) = 10*a(n-1) - 17*a(n-2) for n > 1; a(0) = 1, a(1) = 11.
%F a(n) = ((2+3*sqrt(2))*(5+2*sqrt(2))^n + (2-3*sqrt(2))*(5-2*sqrt(2))^n)/4.
%F G.f.: (1+x)/(1 - 10*x + 17*x^2).
%F a(n) = (17)^((n-1)/2)*(sqrt(17)*ChebyshevU(n, 5/sqrt(17)) + ChebyshevU(n-1, 5/sqrt(17))). - _G. C. Greubel_, Jul 17 2021
%t LinearRecurrence[{10,-17},{1,11},30] (* _Harvey P. Dale_, Jun 04 2012 *)
%o (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((2+3*r)*(5+2*r)^n+(2-3*r)*(5-2*r)^n)/4: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Aug 19 2009
%o (Sage) [(17)^((n-1)/2)*(sqrt(17)*chebyshev_U(n, 5/sqrt(17)) + chebyshev_U(n-1, 5/sqrt(17))) for n in (0..30)] # _G. C. Greubel_, Jul 17 2021
%Y Cf. A164546, A164640.
%K nonn,easy
%O 0,2
%A Al Hakanson (hawkuu(AT)gmail.com), Aug 15 2009
%E Edited and extended beyond a(5) by _Klaus Brockhaus_, Aug 19 2009