A163350 - OEIS

%I #14 Sep 08 2022 08:45:46

%S 1,6,34,188,1028,5592,30344,164464,890896,4824672,26124832,141453248,

%T 765878336,4146681216,22451153024,121555687168,658129355008,

%U 3563255219712,19292230787584,104452273224704,565526954771456

%N a(n) = 8*a(n-1) - 14*a(n-2) for n > 1; a(0) = 1, a(1) = 6.

%C Binomial transform of A102285. Fourth binomial transform of A163403. Inverse binomial transform of A163346.

%H Harvey P. Dale, <a href="/A163350/b163350.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 (8, -14).

%F a(n) = 8*a(n-1) - 14*a(n-2) for n > 1; a(0) = 1, a(1) = 6.

%F a(n) = ((1+sqrt(2))*(4+sqrt(2))^n+(1-sqrt(2))*(4-sqrt(2))^n)/2.

%F G.f.: (1-2*x)/(1-8*x+14*x^2).

%F E.g.f.: exp(4*x)*( cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x) ). - _G. C. Greubel_, Dec 19 2016

%t LinearRecurrence[{8,-14},{1,6},30] (* _Harvey P. Dale_, May 08 2014 *)

%o (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((1+r)*(4+r)^n+(1-r)*(4-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Jul 26 2009

%o (PARI) Vec((1-2*x)/(1-8*x+14*x^2) + O(x^50)) \\ _G. C. Greubel_, Dec 19 2016

%Y Cf. A102285, A163403, A163346.

%K nonn,easy

%O 0,2

%A Al Hakanson (hawkuu(AT)gmail.com), Jul 25 2009

%E Edited and extended beyond a(5) by _Klaus Brockhaus_, Jul 26 2009

%E New name from _G. C. Greubel_, Dec 19 2016