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A083878 a(0)=1, a(1)=3, a(n) = 6*a(n-1) - 7*a(n-2), n >= 2.

%I A083878 #30 May 06 2025 10:33:18
%S A083878 1,3,11,45,193,843,3707,16341,72097,318195,1404491,6199581,27366049,
%T A083878 120799227,533233019,2353803525,10390190017,45864515427,202455762443,
%U A083878 893682966669,3944907462913,17413664010795,76867631824379
%N A083878 a(0)=1, a(1)=3, a(n) = 6*a(n-1) - 7*a(n-2), n >= 2.
%C A083878 Binomial transform of A006012. Second binomial transform of A001333.
%C A083878 Third binomial transform of A077957. Inverse binomial transform of A083879. - _Philippe Deléham_, Dec 01 2008
%H A083878 Michael De Vlieger, <a href="/A083878/b083878.txt">Table of n, a(n) for n = 0..1551</a>
%H A083878 Yassine Otmani, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL28/Otmani/otmani10.html">The 2-Pascal Triangle and a Related Riordan Array</a>, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 12.
%H A083878 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-7).
%F A083878 a(n) = ((3 - sqrt(2))^n + (3 + sqrt(2))^n)/2;
%F A083878 a(n) = Sum_{k=0..n} C(n, 2k)*3^(n-2k)*2^k;
%F A083878 G.f.: (1-3x)/(1-6x+7x^2);
%F A083878 E.g.f.: exp(3x)*cosh(x*sqrt(2)).
%F A083878 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 A083878 a(n) = Sum_{k=0..n} A098158(n,k)*3^(2k-n)*2^(n-k). - _Philippe Deléham_, Dec 01 2008
%F A083878 a(n) = A081179(n+1) - 3*A081179(n). - _R. J. Mathar_, Nov 10 2013
%F A083878 a(n) = Sum_{k=1..n} A056241(n, k) * 2^(k-1). - _J. Conrad_, Nov 23 2022
%t A083878 f[n_] := Simplify[(3 + Sqrt@2)^n + (3 - Sqrt@2)^n]/2; Array[f, 23, 0] (* _Robert G. Wilson v_, Oct 31 2010 *)
%Y A083878 Cf. A083879, A056241, A081179, A098158.
%K A083878 easy,nonn
%O A083878 0,2
%A A083878 _Paul Barry_, May 08 2003