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%I #22 Jan 01 2024 11:44:42
%S 1,1,15,43,281,1121,5895,26363,129361,601441,2884575,13587883,
%T 64675241,305992961,1452764055,6883436603,32652805921,154790287681,
%U 734067052335,3480407844523,16503687369401,78252676717601
%N a(n) = 2*a(n-1) + 13*a(n-2), for n>1, a(0)=1, a(1)=1.
%C Binomial transform of A001023 (powers of 14), with interpolated zeros.
%C a(n) is the number of compositions of n when there are 1 type of 1 and 14 types of other natural numbers. - _Milan Janjic_, Aug 13 2010
%H G. C. Greubel, <a href="/A133343/b133343.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 (2,13).
%F G.f.: (1-x)/(1-2*x-13*x^2).
%F a(n) = Sum_{k=0..n} A098158(n,k)*14^(n-k). - _Philippe Deléham_, Dec 26 2007
%F If p[1]=1, and p[i]=14, (i>1), and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det A. - _Milan Janjic_, Apr 29 2010
%F a(n) = (b*i)^(n-1)*(b*i*ChebyshevU(n, -i/b) - ChebyshevU(n-1, -i/b)), with b = sqrt(13). - _G. C. Greubel_, Oct 15 2022
%t f[n_]:= Simplify[((1+Sqrt[14])^n + (1-Sqrt[14])^n)/2]; Array[f, 25, 0] (* Or *)
%t CoefficientList[Series[(1+13x)/(1-2x-13x^2), {x, 0, 23}], x] (* Or *)
%t LinearRecurrence[{2, 13}, {1, 1}, 25] (* Or *)
%t Table[ MatrixPower[{{1, 2}, {7, 1}}, n][[1, 1]], {n, 0, 30}] (* _Robert G. Wilson v_, Sep 18 2013 *)
%o (PARI) Vec((1-x)/(1-2*x-13*x^2)+O(x^99)) \\ _Charles R Greathouse IV_, Jan 12 2012
%o (Magma) [n le 2 select 1 else 2*Self(n-1) +13*Self(n-2): n in [1..41]]; // _G. C. Greubel_, Oct 15 2022
%o (SageMath)
%o A133343=BinaryRecurrenceSequence(2,13,1,1)
%o [A133343(n) for n in range(41)] # _G. C. Greubel_, Oct 15 2022
%Y Cf. A001023, A098158.
%K nonn,easy
%O 0,3
%A _Philippe Deléham_, Dec 21 2007
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