A147960 - OEIS

%I #22 Sep 08 2022 08:45:38

%S 1,9,83,783,7537,73809,733139,7365591,74662657,762046137,7818480563,

%T 80531005311,831898131121,8612216940609,89299952572403,

%U 927034007995143,9631915890692737,100138799400852969,1041577033850627219

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

%C Binomial transform of A147959. 9th binomial transform of A077957. - _Philippe Deléham_, Nov 30 2008

%C Hankel transform is := [1, 2, 0, 0, 0, 0, 0, 0, 0, 0, ...]. - _Philippe Deléham_, Dec 04 2008

%H G. C. Greubel, <a href="/A147960/b147960.txt">Table of n, a(n) for n = 0..980</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (18, -79).

%F From _Philippe Deléham_, Nov 19 2008: (Start)

%F a(n) = 18*a(n-1) - 79*a(n-2), n > 1; a(0)=1, a(1)=9.

%F G.f.: (1 - 9*x)/(1 - 18*x + 79*x^2).

%F a(n) = (Sum_{k=0..n} A098158(n,k)*9^(2k)*2^(n-k))/9^n. (End)

%F E.g.f.: exp(9*x)*cosh(sqrt(2)*x). - _Ilya Gutkovskiy_, Aug 11 2017

%t LinearRecurrence[{18, -79}, {1, 9}, 50] (* _G. C. Greubel_, Aug 17 2018 *)

%o (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 (PARI) x='x+O('x^30); Vec((1-9*x)/(1-18*x+79*x^2)) \\ _G. C. Greubel_, Aug 17 2018

%Y Cf. A077957, A098158, A147959.

%K nonn

%O 0,2

%A Al Hakanson (hawkuu(AT)gmail.com), Nov 17 2008

%E Extended beyond a(6) by _Klaus Brockhaus_, Nov 19 2008