A081342 - OEIS

%I #28 Sep 08 2022 08:45:09

%S 1,5,34,260,2056,16400,131104,1048640,8388736,67109120,536871424,

%T 4294968320,34359740416,274877911040,2199023263744,17592186060800,

%U 140737488388096,1125899906908160,9007199254872064,72057594038190080,576460752303947776,4611686018428436480

%N a(n) = (8^n + 2^n)/2.

%C Binomial transform of A034494.

%C 5th binomial transform of {1, 0, 9, 0, 81, 0, 729, 0, ...}.

%H Vincenzo Librandi, <a href="/A081342/b081342.txt">Table of n, a(n) for n = 0..200</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10,-16).

%F a(n) = (8^n + 2^n)/2.

%F a(n) = 10*a(n-1) - 16*a(n-2), a(0)=1, a(1)=5.

%F G.f.: (1-5*x)/((1-2*x)*(1-8*x)).

%F E.g.f.: exp(5*x)*cosh(3*x).

%F a(n) = ((5+sqrt(9))^n + (5-sqrt(9))^n)/2. - Al Hakanson (hawkuu(AT)gmail.com), Dec 08 2008

%F a(n) = A074603(n)/2. - _Michel Marcus_, Jan 09 2020

%p seq( (8^n + 2^n)/2, n=0..30); # _G. C. Greubel_, Jan 08 2020

%t Table[(8^n + 2^n)/2, {n, 0, 30}] (* _Vladimir Joseph Stephan Orlovsky_, Jun 12 2011 *)

%o (Magma) [(8^n+2^n)/2: n in [0..30]]; // _Vincenzo Librandi_, Jun 13 2011

%o (PARI) a(n)=(8^n+2^n)/2 \\ _Charles R Greathouse IV_, Sep 28 2015

%o (Sage) [(8^n + 2^n)/2 for n in (0..30)] # _G. C. Greubel_, Jan 08 2020

%o (GAP) List([0..30], n-> (8^n + 2^n)/2); # _G. C. Greubel_, Jan 08 2020

%Y Cf. A074603, A025551, A081343.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Mar 18 2003