A090042 a(n) = 2*a(n-1) + 11*a(n-2) for n > 1, a(0) = a(1) = 1.

1, 1, 13, 37, 217, 841, 4069, 17389, 79537, 350353, 1575613, 7005109, 31341961, 139740121, 624241813, 2785624957, 12437909857, 55517694241, 247852396909, 1106399430469, 4939175226937, 22048744189033, 98428415874373, 439393017828109, 1961498610274321, 8756320416657841

Offset: 0

Author

Paul Barry, Nov 20 2003

Keywords

• nonn
• easy

Comments

Binomial transform of A001021 (powers of 12), with interpolated zeros.

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

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

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..319
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (2,11).

Crossrefs

Cf. A049310, A015520, A084603.

Programs

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

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

Maple

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:
seq(a(n), n=0..25); # Muniru A Asiru, Feb 18 2018

Mathematica

a[n_]:= Simplify[((1+Sqrt[12])^n +(1-Sqrt[12])^n)/2]; Array[a, 30, 0] (* or *)
CoefficientList[Series[(x-1)/(11x^2+2x-1), {x,0,30}], x] (* or *)
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 *)
LinearRecurrence[{2, 11}, {1, 1}, 30] (* Ray Chandler, Aug 01 2015 *)

PARI

x='x+O('x^30); Vec((1-x)/(1-2*x-11*x^2)) \\ Altug Alkan, Feb 17 2018

Sage

((1-x)/(1-2*x-11*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 02 2019

Formula

E.g.f.: exp(x)*cosh(2*sqrt(3)*x).

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

a(n) = Sum_{k=0..n} A098158(n,k)*12^(n-k). - Philippe Deléham, Dec 26 2007

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

From Wolfdieter Lang, Feb 17 2018: (Start)

G.f.: (1-x)/(1 - 2*x - 11*x^2). (See the Mathematica program.)

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.

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)

From Peter Bala, Jan 07 2022: (Start)

a(n) = [x^n] (x + sqrt(1 + 12*x^2))^n.

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.

O.g.f.: 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)