%I #20 Sep 08 2022 08:45:40
%S 1,14,153,1540,14981,143514,1365013,12939080,122451561,1157941414,
%T 10945762673,103449196620,977620957741,9238377953714,87299590169133,
%U 824944010358160,7795333767741521,73662080302980414,696069772228840393
%N a(n) = ( (7 + sqrt(6))^n - (7 - sqrt(6))^n )/(2*sqrt(6)).
%C lim_{n -> infinity} a(n)/a(n-1) = 7 + sqrt(6) = 9.4494897427....
%H G. C. Greubel, <a href="/A154239/b154239.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (14, -43).
%F From _Philippe Deléham_, Jan 06 2009: (Start)
%F a(n) = 14*a(n-1) - 43*a(n-2)for n>1, with a(0)=0, a(1)=1.
%F G.f.: x/(1 - 14x + 43x^2). (End)
%F E.g.f.: sinh(sqrt(6)*x)*exp(7*x)/sqrt(6). - _Ilya Gutkovskiy_, Sep 07 2016
%t Join[{a=1,b=14},Table[c=14*b-43*a;a=b;b=c,{n,60}]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 01 2011 *)
%t LinearRecurrence[{14, -43}, {1, 14}, 25] (* or *) Table[( (7 + sqrt(6))^n - (7 - sqrt(6))^n )/(2*sqrt(6)), {n,1,25}] (* _G. C. Greubel_, Sep 07 2016 *)
%o (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-6); S:=[ ((7+r)^n-(7-r)^n)/(2*r): n in [1..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Jan 07 2009
%o I:=[1,14]; [n le 2 select I[n] else 14*Self(n-1)-43*Self(n-2): n in [1..20]]; // _Vincenzo Librandi_, Sep 07 2016
%Y Cf. A010464 (decimal expansion of square root of 6).
%K nonn
%O 1,2
%A Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009
%E Extended beyond a(7) by _Klaus Brockhaus_, Jan 07 2009
%E Edited by _Klaus Brockhaus_, Oct 06 2009