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a(n) = (2*5^n + (-1)^n)/3.
2

%I #26 Feb 18 2023 10:57:05

%S 1,3,17,83,417,2083,10417,52083,260417,1302083,6510417,32552083,

%T 162760417,813802083,4069010417,20345052083,101725260417,508626302083,

%U 2543131510417,12715657552083,63578287760417,317891438802083

%N a(n) = (2*5^n + (-1)^n)/3.

%C Binomial transform of A003683 (without leading zero). Inverse binomial transform of A067411.

%C a(n) is the number of compositions of n when there are 3 types of 1 and 8 types of other natural numbers. - _Milan Janjic_, Aug 13 2010

%H G. C. Greubel, <a href="/A083217/b083217.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,5).

%F a(n) = (2*5^n + (-1)^n)/3.

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

%F E.g.f.: (2*exp(5*x) + exp(-x))/3

%F a(n) = Sum_{k=0..n} Sum_{j=0..n-k} C(n,j)*C(n-j,k)*J(n-j+1) where J(n) = A001045(n). - _Paul Barry_, May 19 2006

%F a(0)=1, a(n) = 5*a(n-1) - 2 if n is odd, and a(n) = 5*a(n) + 2 if n is even. - _Vincenzo Librandi_, Nov 18 2010

%t LinearRecurrence[{4,5},{1,3},30] (* _Harvey P. Dale_, Sep 18 2018 *)

%o (Sage)

%o from sage.combinat.sloane_functions import recur_gen2b

%o it = recur_gen2b(1,3,4,5, lambda n: 0)

%o [next(it) for i in range(1,24)] # _Zerinvary Lajos_, Jul 03 2008

%o (Magma) [(2*5^n +(-1)^n)/3: n in [0..40]]; // _G. C. Greubel_, Feb 17 2023

%Y Cf. A003683, A067411, A082412.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Apr 23 2003