A161158 a(n) = A003696(n+1)/A001353(n+1).

1, 14, 161, 1792, 19809, 218638, 2412353, 26614784, 293628097, 3239445006, 35739069409, 394290020096, 4349990523425, 47991114171406, 529460241815169, 5841251080892416, 64443392518654337, 710969410782059534

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

Cf. A001563, A003696, A100047.

Sequence in context: A282043 A193103 A016206 * A238116 A153664 A016146

Adjacent sequences: A161155 A161156 A161157 * A161159 A161160 A161161

Keyword

nonn,easy

Author

R. J. Mathar, Jun 03 2009

Status

approved