A164609 - OEIS

%I #15 Sep 08 2022 08:45:47

%S 1,13,113,909,7169,56237,440497,3448941,27000961,211377613,1654759793,

%T 12954178509,101410868609,793887651437,6214891748017,48652827405741,

%U 380875114341121,2981653077513613,23341653831337073,182728435995639309

%N a(n) = 10*a(n-1) - 17*a(n-2) for n > 1; a(0) = 1, a(1) = 13.

%C Binomial transform of A164608. Fifth binomial transform of A164683.

%H G. C. Greubel, <a href="/A164609/b164609.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..144 from Vincenzo Librandi)

%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) = 13.

%F a(n) = ((2+4*sqrt(2))*(5+2*sqrt(2))^n + (2-4*sqrt(2))*(5-2*sqrt(2))^n)/4.

%F G.f.: (1+3*x)/(1-10*x+17*x^2).

%F E.g.f.: exp(5*x)*(cosh(2*sqrt(2)*x) + 2*sqrt(2)*sinh(2*sqrt(2)*x)). - _G. C. Greubel_, Aug 10 2017

%t LinearRecurrence[{10,-17},{1,13},20] (* _Harvey P. Dale_, Nov 05 2014 *)

%o (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((2+4*r)*(5+2*r)^n+(2-4*r)*(5-2*r)^n)/4: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Aug 22 2009

%o (PARI) x='x+O('x^50); Vec((1+3*x)/(1-10*x+17*x^2)) \\ _G. C. Greubel_, Aug 10 2017

%Y Cf. A164608, A164683.

%K nonn,easy

%O 0,2

%A Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009

%E Edited and extended beyond a(5) by _Klaus Brockhaus_, Aug 22 2009