%I #31 Sep 08 2022 08:45:40
%S 1,18,245,2988,34429,383670,4186169,45041112,480032665,5082340122,
%T 53559541661,562566880260,5895000053461,61667217421758,
%U 644304909368225,6725778192309168,70163919621475249,731614075994130210
%N a(n) = ((9 + sqrt(2))^n - (9 - sqrt(2))^n)/(2*sqrt(2)).
%C Preceded by zero, this is the eighth binomial transform of the Pell sequence A000129. - _Sergio Falcon_, Sep 21 2009; edited by _Klaus Brockhaus_, Oct 11 2009
%C Eighth binomial transform of A048697.
%C First differences are in A164600.
%C lim_{n -> infinity} a(n)/a(n-1) = 9 + sqrt(2) = 10.4142135623....
%H Vincenzo Librandi, <a href="/A153593/b153593.txt">Table of n, a(n) for n = 1..100</a>
%H S. Falcon, <a href="http://dx.doi.org/10.9734/BJMCS/2014/11783">Iterated Binomial Transforms of the k-Fibonacci Sequence</a>, British Journal of Mathematics & Computer Science, 4 (22): 2014.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (18,-79).
%F a(n) = 18*a(n-1) - 79*a(n-2) for n>1; a(0)=0, a(1)=1. - _Philippe Deléham_, Jan 01 2009
%F G.f.: x/(1 - 18*x + 79*x^2). - _Klaus Brockhaus_, Dec 31 2008, corrected Oct 11 2009
%F a(n) = Sum[Binomial[n - 1 - i, i] (-1)^i * 18^(n - 1 - 2 i) * 79^i, {i, 0, Floor[(n - 1)/2]}]. - _Sergio Falcon_, Sep 21 2009
%F E.g.f.: exp(9*x)*sinh(sqrt(2)*x)/sqrt(2). - _Ilya Gutkovskiy_, Aug 12 2017
%t Join[{a=1,b=18},Table[c=18*b-79*a;a=b;b=c,{n,40}]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 09 2011 *)
%t LinearRecurrence[{18,-79},{1,18},25] (* _G. C. Greubel_, Aug 22 2016 *)
%o (Magma) Z<x>:= PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((9+r)^n-(9-r)^n)/(2*r): n in [1..18] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Dec 31 2008
%Y Cf. A000129 (Pell numbers), A007070, A081185, A081184, A081183, A081182, A081180, A081179. - _Sergio Falcon_, Sep 21 2009
%Y Cf. A002193 (decimal expansion of sqrt(2)), A048697, A164600.
%K nonn
%O 1,2
%A Al Hakanson (hawkuu(AT)gmail.com), Dec 29 2008
%E Extended beyond a(7) by _Klaus Brockhaus_, Dec 31 2008
%E Edited by _Klaus Brockhaus_, Oct 11 2009