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A146964 a(n) = ((4 + sqrt(7))^n + (4 - sqrt(7))^n)/2.

%I A146964 #20 Dec 28 2023 19:31:46
%S A146964 1,4,23,148,977,6484,43079,286276,1902497,12643492,84025463,558412276,
%T A146964 3711069041,24662841844,163903113383,1089259330468,7238946623297,
%U A146964 48108239012164,319715392487639,2124748988791636,14120553377944337
%N A146964 a(n) = ((4 + sqrt(7))^n + (4 - sqrt(7))^n)/2.
%C A146964 Binomial transform of A146963.
%C A146964 Inverse binomial transform of A146965.
%H A146964 Vincenzo Librandi, <a href="/A146964/b146964.txt">Table of n, a(n) for n = 0..300</a>
%H A146964 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-9).
%F A146964 From _Philippe Deléham_ and _Klaus Brockhaus_, Nov 05 2008: (Start)
%F A146964 a(n) = 8*a(n-1) - 9*a(n-2) with a(0)=1, a(1)=4.
%F A146964 G.f.: (1-4*x)/(1-8*x+9*x^2). (End)
%F A146964 a(n) = (Sum_{k=0..n} A098158(n,k)*4^(2*k)*7^(n-k))/4^n. - _Philippe Deléham_, Nov 06 2008
%F A146964 E.g.f.: exp(4*x)*cosh(sqrt(7)*x). - _G. C. Greubel_, Jan 08 2020
%p A146964 seq(coeff(series((1-4*x)/(1-8*x+9*x^2), x, n+1), x, n), n = 0..25); # _G. C. Greubel_, Jan 08 2020
%t A146964 LinearRecurrence[{8,-9}, {1,4}, 25] (* _G. C. Greubel_, Jan 08 2020 *)
%o A146964 (Magma) Z<x>:= PolynomialRing(Integers()); N<r7>:=NumberField(x^2-7); S:=[ ((4+r7)^n+(4-r7)^n)/2: n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Nov 05 2008
%o A146964 (PARI) my(x='x+O('x^25)); Vec((1-4*x)/(1-8*x+9*x^2)) \\ _G. C. Greubel_, Jan 08 2020
%o A146964 (Sage)
%o A146964 def A146964_list(prec):
%o A146964     P.<x> = PowerSeriesRing(ZZ, prec)
%o A146964     return P( (1-4*x)/(1-8*x+9*x^2) ).list()
%o A146964 A146964_list(25) # _G. C. Greubel_, Jan 08 2020
%o A146964 (GAP) a:=[1,4];; for n in [3..25] do a[n]:=8*a[n-1]-9*a[n-2]; od; a; # _G. C. Greubel_, Jan 08 2020
%Y A146964 Cf. A098158, A146963, A146965.
%K A146964 nonn
%O A146964 0,2
%A A146964 Al Hakanson (hawkuu(AT)gmail.com), Nov 03 2008
%E A146964 Extended beyond a(7) by _Klaus Brockhaus_, Nov 05 2008
%E A146964 Edited by _Klaus Brockhaus_, Jul 16 2009