OFFSET
0,2
COMMENTS
Proposed by R. Guy in the seqfan list Mar 28 2009.
With an offset of 1, this sequence is the case P1 = 14, P2 = 32, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Apr 27 2014
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..900
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume.
Index entries for linear recurrences with constant coefficients, signature (14,-34,14,-1).
FORMULA
a(n) = 14*a(n-1) -34*a(n-2) +14*a(n-3) -a(n-4).
G.f.: (1-x^2)/(1-14*x+34*x^2-14*x^3+x^4).
From Peter Bala, Apr 27 2014: (Start)
The following remarks assume an offset of 1.
a(n) = (1/sqrt(17))*( T(n,(7 + sqrt(17))/2) - T(n,(7 - sqrt(17))/2) ), where T(n,x) is the Chebyshev polynomial of the first kind.
a(n) = the bottom left entry of the 2 X 2 matrix T(n,M), where M is the 2 X 2 matrix [0, -8; 1, 7].
a(n) = U(n-1,1/2*(4 + sqrt(2)))*U(n-1,1/2*(4 - sqrt(2))), where U(n,x) is the Chebyshev polynomial of the second kind.
See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4th-order linear divisibility sequences. (End)
MAPLE
seq(simplify( ChebyshevU(n, (4+sqrt(2))/2)*ChebyshevU(n, (4-sqrt(2))/2) ), n = 0 .. 20); # G. C. Greubel, Dec 24 2019
MATHEMATICA
CoefficientList[Series[(1-x^2)/(1-14x+34x^2-14x^3+x^4), {x, 0, 20}], x] (* Vincenzo Librandi, Apr 28 2014 *)
Table[Simplify[ChebyshevU[n, (4+Sqrt[2])/2]*ChebyshevU[n, (4-Sqrt[2])/2]], {n, 0, 20}] (* G. C. Greubel, Dec 24 2019 *)
PROG
(Magma) I:=[1, 14, 161, 1792]; [n le 4 select I[n] else 14*Self(n-1)-34*Self(n-2) +14*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Apr 28 2014
(PARI) vector(21, n, round(polchebyshev(n-1, 2, (4+sqrt(2))/2)*polchebyshev(n-1, 2, (4-sqrt(2))/2)) ) \\ G. C. Greubel, Dec 24 2019
(Sage) [round(chebyshev_U(n, (4+sqrt(2))/2)*chebyshev_U(n, (4-sqrt(2))/2)) for n in (0..20)] # G. C. Greubel, Dec 24 2019
(GAP) a:=[1, 14, 161, 1792];; for n in [5..20] do a[n]:=14*a[n-1]-34*a[n-2] +14*a[n-3] -a[n-4]; od; a; # G. C. Greubel, Dec 24 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
R. J. Mathar, Jun 03 2009
STATUS
approved