cp's OEIS Frontend

A083217 a(n) = (2*5^n + (-1)^n)/3.

%I A083217 #26 Feb 18 2023 10:57:05
%S A083217 1,3,17,83,417,2083,10417,52083,260417,1302083,6510417,32552083,
%T A083217 162760417,813802083,4069010417,20345052083,101725260417,508626302083,
%U A083217 2543131510417,12715657552083,63578287760417,317891438802083
%N A083217 a(n) = (2*5^n + (-1)^n)/3.
%C A083217 Binomial transform of A003683 (without leading zero). Inverse binomial transform of A067411.
%C A083217 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 A083217 G. C. Greubel, <a href="/A083217/b083217.txt">Table of n, a(n) for n = 0..1000</a>
%H A083217 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,5).
%F A083217 a(n) = (2*5^n + (-1)^n)/3.
%F A083217 G.f.: (1-x)/((1-5*x)*(1+x)).
%F A083217 E.g.f.: (2*exp(5*x) + exp(-x))/3
%F A083217 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 A083217 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 A083217 LinearRecurrence[{4,5},{1,3},30] (* _Harvey P. Dale_, Sep 18 2018 *)
%o A083217 (Sage)
%o A083217 from sage.combinat.sloane_functions import recur_gen2b
%o A083217 it = recur_gen2b(1,3,4,5, lambda n: 0)
%o A083217 [next(it) for i in range(1,24)] # _Zerinvary Lajos_, Jul 03 2008
%o A083217 (Magma) [(2*5^n +(-1)^n)/3: n in [0..40]]; // _G. C. Greubel_, Feb 17 2023
%Y A083217 Cf. A003683, A067411, A082412.
%K A083217 easy,nonn
%O A083217 0,2
%A A083217 _Paul Barry_, Apr 23 2003