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%I #57 Sep 08 2022 08:45:12
%S 1,1,13,37,217,841,4069,17389,79537,350353,1575613,7005109,31341961,
%T 139740121,624241813,2785624957,12437909857,55517694241,247852396909,
%U 1106399430469,4939175226937,22048744189033,98428415874373,439393017828109,1961498610274321,8756320416657841
%N a(n) = 2*a(n-1) + 11*a(n-2) for n > 1, a(0) = a(1) = 1.
%C Binomial transform of A001021 (powers of 12), with interpolated zeros.
%C For n > 0, a(n) = term (1,1) in the n-th power of the 2 X 2 matrix [1,3; 4,1]. - _Gary W. Adamson_, Aug 06 2010
%C a(n) is the number of compositions of n when there are 1 type of 1 and 12 types of other natural numbers. - _Milan Janjic_, Aug 13 2010
%H Muniru A Asiru, <a href="/A090042/b090042.txt">Table of n, a(n) for n = 0..319</a>
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,11).
%F E.g.f.: exp(x)*cosh(2*sqrt(3)*x).
%F a(n) = ((1 + 2*sqrt(3))^n + (1 - 2*sqrt(3))^n)/2.
%F a(n) = Sum_{k=0..n} A098158(n,k)*12^(n-k). - _Philippe Deléham_, Dec 26 2007
%F If p[1]=1, and p[i]=12, (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 From _Wolfdieter Lang_, Feb 17 2018: (Start)
%F G.f.: (1-x)/(1 - 2*x - 11*x^2). (See the Mathematica program.)
%F a(n) = b(n+1) - b(n), with b(n) = A015520(n). This leads to the Binet-de Moivre type formula given in the Mathematica program.
%F a(n) = (i*sqrt(11))^n*(S(n,-2*i/sqrt(11)) + (i/sqrt(11))*S(n-1,-2*i/sqrt(11))), n >= 0, with Chebyshev S polynomials (coefficients in A049310), with S(-2, x) = -1, S(-1, x) = 0 and i = sqrt(-1). Via Cayley-Hamilton. See the _Gary W. Adamson_ comment above or the Mathematica program of _Robert G. Wilson v_ with another matrix. (End)
%F From _Peter Bala_, Jan 07 2022: (Start)
%F a(n) = [x^n] (x + sqrt(1 + 12*x^2))^n.
%F The Gauss congruences a(n*p^k) == a(n^p^(k-1)) (mod p^k) hold for prime p and positive integers n and k.
%F O.g.f. A(x) = 1 + x*d/dx(log(B(x)), where B(x) = 1/sqrt(1 - 2*x - 11*x^2) is the o.g.f. of A084603. (End)
%p a := proc(n) option remember: if n=0 then 1 elif n=1 then 1 elif n>=2 then 2*procname(n-1) + 11*procname(n-2) fi; end:
%p seq(a(n), n=0..25); # _Muniru A Asiru_, Feb 18 2018
%t a[n_]:= Simplify[((1+Sqrt[12])^n +(1-Sqrt[12])^n)/2]; Array[a, 30, 0] (* or *)
%t CoefficientList[Series[(x-1)/(11x^2+2x-1), {x,0,30}], x] (* or *)
%t Table[ MatrixPower[{{1, 2}, {6, 1}}, n][[1, 1]], {n, 0, 30}] (* _Robert G. Wilson v_, Sep 18 2013 and modified per _Wolfdieter Lang_ Feb 17 2018 *)
%t LinearRecurrence[{2, 11}, {1, 1}, 30] (* _Ray Chandler_, Aug 01 2015 *)
%o (PARI) x='x+O('x^30); Vec((1-x)/(1-2*x-11*x^2)) \\ _Altug Alkan_, Feb 17 2018
%o (GAP) a := [1, 1];; for n in [3..30] do a[n] := 2*a[n-1]+ 11*a[n-2]; od; a; # _Muniru A Asiru_, Feb 18 2018
%o (Magma) I:=[1,1]; [n le 2 select I[n] else 2*Self(n-1) +11*Self(n-2): n in [1..30]]; // _G. C. Greubel_, Aug 02 2019
%o (Sage) ((1-x)/(1-2*x-11*x^2)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Aug 02 2019
%Y Cf. A049310, A015520, A084603.
%K nonn,easy
%O 0,3
%A _Paul Barry_, Nov 20 2003
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