A083878 - OEIS

%I #26 Dec 25 2022 17:36:24

%S 1,3,11,45,193,843,3707,16341,72097,318195,1404491,6199581,27366049,

%T 120799227,533233019,2353803525,10390190017,45864515427,202455762443,

%U 893682966669,3944907462913,17413664010795,76867631824379

%N a(0)=1, a(1)=3, a(n) = 6*a(n-1) - 7*a(n-2), n >= 2.

%C Binomial transform of A006012. Second binomial transform of A001333.

%C Third binomial transform of A077957. Inverse binomial transform of A083879. - _Philippe Deléham_, Dec 01 2008

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-7).

%F a(n) = ((3 - sqrt(2))^n + (3 + sqrt(2))^n)/2;

%F a(n) = Sum_{k=0..n} C(n, 2k)*3^(n-2k)*2^k;

%F G.f.: (1-3x)/(1-6x+7x^2);

%F E.g.f.: exp(3x)*cosh(x*sqrt(2)).

%F a(n) = Sum_{k=0..n} C(n, k)*2^((n-k)/2)(1+(-1)^(n-k))*3^k/2. - _Paul Barry_, Jan 22 2005

%F a(n) = Sum_{k=0..n} A098158(n,k)*3^(2k-n)*2^(n-k). - _Philippe Deléham_, Dec 01 2008

%F a(n) = A081179(n+1) - 3*A081179(n). - _R. J. Mathar_, Nov 10 2013

%F a(n) = Sum_{k=1..n} A056241(n, k) * 2^(k-1). - _J. Conrad_, Nov 23 2022

%t f[n_] := Simplify[(3 + Sqrt@2)^n + (3 - Sqrt@2)^n]/2; Array[f, 23, 0] (* _Robert G. Wilson v_, Oct 31 2010 *)

%Y Cf. A083879, A056241, A081179, A098158.

%K easy,nonn

%O 0,2

%A _Paul Barry_, May 08 2003