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

%I #17 Sep 08 2022 08:45:46

%S 5,14,46,156,532,1816,6200,21168,72272,246752,842464,2876352,9820480,

%T 33529216,114475904,390845184,1334428928,4556025344,15555243520,

%U 53108923392,181325206528,619082979328,2113681504256,7216560058368

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

%C Binomial transform of A163607. Inverse binomial transform of A163609.

%H Harvey P. Dale, <a href="/A163608/b163608.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2).

%F a(n) = 4*a(n-1) - 2*a(n-2) for n > 1; a(0) = 5, a(1) = 14.

%F G.f.: (5-6*x)/(1-4*x+2*x^2).

%F a(n) = 5*A007070(n) - 6*A007070(n-1). - _R. J. Mathar_, Nov 08 2013

%F E.g.f.: exp(2*x)*( 5*cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x) ). - _G. C. Greubel_, Jul 29 2017

%t LinearRecurrence[{4,-2},{5,14},30] (* _Harvey P. Dale_, Jan 31 2017 *)

%o (Magma) Z<x>:= PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((5+2*r)*(2+r)^n+(5-2*r)*(2-r)^n)/2: n in [0..23] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Aug 06 2009

%o (PARI) x='x+O('x^50); Vec((5-6*x)/(1-4*x+2*x^2)) \\ _G. C. Greubel_, Jul 29 2017

%Y Cf. A163607, A163609.

%K nonn,easy

%O 0,1

%A Al Hakanson (hawkuu(AT)gmail.com), Aug 01 2009

%E Edited and extended beyond a(5) by _Klaus Brockhaus_, Aug 06 2009