OFFSET
0,2
COMMENTS
7*a(n)^2 - 5*b(n)^2 = 2 with companion sequence b(n) =A077416(n), n>=0.
a(n) = L(n,12), where L is defined as inA108299;see alsoA077416for L(n,-12). -Reinhard Zumkeller,Jun 01 2005
[a(n),A004191(n)] = the 2 X 2 matrix [1,10; 1,11]^(n+1) * [1,0]. -Gary W. Adamson,Mar 19 2008
Hankel transform ofA174227.-Paul Barry,Mar 12 2010
Alternate denominators of the continued fraction convergents to sqrt(35), seeA041059.-James R. Buddenhagen,May 20 2010
For positive n, a(n) equals the permanent of the (2n)X(2n) tridiagonal matrix with sqrt(10)'s along the main diagonal, and 1's along the superdiagonal and the subdiagonal. -John M. Campbell,Jul 08 2011
Positive values of x (or y) satisfying x^2 - 12xy + y^2 + 10 = 0. -Colin Barker,Feb 09 2014
a(n) = a(-1-n) for all n in Z. -Michael Somos,Jun 29 2019
LINKS
Vincenzo Librandi,Table of n, a(n) for n = 0..200
Alex Fink, Richard K. Guy, and Mark Krusemeyer,Partitions with parts occurring at most thrice,Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114. See Section 13.
Tanya Khovanova,Recursive Sequences
J.-C. Novelli, J.-Y. Thibon,Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions,arXiv:1403.5962 [math.CO], 2014.
Index entries for linear recurrences with constant coefficients,signature (12,-1).
FORMULA
a(n) = 12*a(n-1) - a(n-2), a(-1)=1, a(0)=1.
a(n) = S(n, 12) - S(n-1, 12) = T(2*n+1, sqrt(14)/2)/(sqrt(14)/2) with S(n, x):= U(n, x/2), resp. T(n, x), Chebyshev's polynomials of the second, resp. first, kind. SeeA049310andA053120.S(-1, x)=0, S(n, 12)=A004191(n).
G.f.: (1-x)/(1-12*x+x^2).
a(n) = (ap^(2*n+1) + am^(2*n+1))/sqrt(14) with ap:= (sqrt(7)+sqrt(5))/sqrt(2) and am:= (sqrt(7)-sqrt(5))/sqrt(2).
a(n) = sqrt((5*A077416(n)^2 + 2)/7).
a(n)*a(n+3) = 120 + a(n+1)*a(n+2). -Ralf Stephan,May 29 2004
EXAMPLE
G.f. = 1 + 11*x + 131*x^2 + 1561*x^3 + 18601*x^4 221651*x^5 + 2641211*x^6 +...
MATHEMATICA
CoefficientList[Series[(1 - x)/(1 - 12 x + x^2), {x, 0, 30}], x] (*Vincenzo Librandi,Feb 10 2014 *)
LinearRecurrence[{12, -1}, {1, 11}, 30] (*Harvey P. Dale,Apr 09 2015 *)
a[ n_]:= With[{x = Sqrt[7/2]}, ChebyshevT[2 n + 1, x]/x] // Expand; (*Michael Somos,Jun 29 2019 *)
PROG
(Magma) I:=[1, 11]; [n le 2 select I[n] else 12*Self(n-1)-Self(n-2): n in [1..30]]; //Vincenzo Librandi,Feb 10 2014
(PARI) x='x+O('x^30); Vec((1-x)/(1-12*x+x^2)) \\G. C. Greubel,Jan 18 2018
(PARI) {a(n) = my(x = quadgen(56)/2); simplify(polchebyshev(2*n + 1, 1, x)/x)}; /*Michael Somos,Jun 29 2019 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang,Nov 29 2002
EXTENSIONS
More terms fromVincenzo Librandi,Feb 10 2014
STATUS
approved