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A147960 a(n) = ((9 + sqrt(2))^n + (9 - sqrt(2))^n)/2.

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%I A147960 #22 Sep 08 2022 08:45:38
%S A147960 1,9,83,783,7537,73809,733139,7365591,74662657,762046137,7818480563,
%T A147960 80531005311,831898131121,8612216940609,89299952572403,
%U A147960 927034007995143,9631915890692737,100138799400852969,1041577033850627219
%N A147960 a(n) = ((9 + sqrt(2))^n + (9 - sqrt(2))^n)/2.
%C A147960 Binomial transform of A147959. 9th binomial transform of A077957. - _Philippe Deléham_, Nov 30 2008
%C A147960 Hankel transform is := [1, 2, 0, 0, 0, 0, 0, 0, 0, 0, ...]. - _Philippe Deléham_, Dec 04 2008
%H A147960 G. C. Greubel, <a href="/A147960/b147960.txt">Table of n, a(n) for n = 0..980</a>
%H A147960 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (18, -79).
%F A147960 From _Philippe Deléham_, Nov 19 2008: (Start)
%F A147960 a(n) = 18*a(n-1) - 79*a(n-2), n > 1; a(0)=1, a(1)=9.
%F A147960 G.f.: (1 - 9*x)/(1 - 18*x + 79*x^2).
%F A147960 a(n) = (Sum_{k=0..n} A098158(n,k)*9^(2k)*2^(n-k))/9^n. (End)
%F A147960 E.g.f.: exp(9*x)*cosh(sqrt(2)*x). - _Ilya Gutkovskiy_, Aug 11 2017
%t A147960 LinearRecurrence[{18, -79}, {1, 9}, 50] (* _G. C. Greubel_, Aug 17 2018 *)
%o A147960 (Magma) Z<x>:= PolynomialRing(Integers()); N<r2>:=NumberField(x^2-2); S:=[ ((9+r2)^n+(9-r2)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Nov 19 2008
%o A147960 (PARI) x='x+O('x^30); Vec((1-9*x)/(1-18*x+79*x^2)) \\ _G. C. Greubel_, Aug 17 2018
%Y A147960 Cf. A077957, A098158, A147959.
%K A147960 nonn
%O A147960 0,2
%A A147960 Al Hakanson (hawkuu(AT)gmail.com), Nov 17 2008
%E A147960 Extended beyond a(6) by _Klaus Brockhaus_, Nov 19 2008