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A147957 a(n) = ((6 + sqrt(2))^n + (6 - sqrt(2))^n)/2.

%I A147957 #27 Oct 28 2023 13:09:54
%S A147957 1,6,38,252,1732,12216,87704,637104,4663312,34298208,253025888,
%T A147957 1870171584,13839178816,102484311936,759279663488,5626889356032,
%U A147957 41707163713792,309171726460416,2292017151256064,16992367115418624,125979822242317312,934017384983574528,6924894663564105728
%N A147957 a(n) = ((6 + sqrt(2))^n + (6 - sqrt(2))^n)/2.
%C A147957 6th binomial transform of A077957. Binomial transform of A083880. Inverse binomial transform of A147958. - _Philippe Deléham_, Nov 30 2008
%H A147957 G. C. Greubel, <a href="/A147957/b147957.txt">Table of n, a(n) for n = 0..1000</a>
%H A147957 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (12,-34).
%F A147957 From _Philippe Deléham_, Nov 19 2008: (Start)
%F A147957 a(n) = 12*a(n-1) - 34*a(n-2), n > 1; a(0)=1, a(1)=6.
%F A147957 G.f.: (1 - 6*x)/(1 - 12*x + 34*x^2).
%F A147957 a(n) = (Sum_{k=0..n} A098158(n,k)*6^(2k)*2^(n-k))/6^n. (End)
%F A147957 E.g.f.: exp(6*x)*cosh(sqrt(2)*x). - _Ilya Gutkovskiy_, Aug 11 2017
%t A147957 LinearRecurrence[{12, -34}, {1, 6}, 50] (* _G. C. Greubel_, Aug 17 2018 *)
%o A147957 (Magma) Z<x>:= PolynomialRing(Integers()); N<r2>:=NumberField(x^2-2); S:=[ ((6+r2)^n+(6-r2)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Nov 19 2008
%o A147957 (PARI) my(x='x+O('x^50)); Vec((1-6*x)/(1-12*x+34*x^2)) \\ _G. C. Greubel_, Aug 17 2018
%Y A147957 Cf. A077957, A083880, A098158, A147958.
%K A147957 nonn,easy
%O A147957 0,2
%A A147957 Al Hakanson (hawkuu(AT)blogspot.com), Nov 17 2008
%E A147957 Extended beyond a(6) by _Klaus Brockhaus_, Nov 19 2008