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

Author

R. J. Mathar, Jun 03 2009

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A001563, A003696, A100047.

Programs

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

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

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 *)

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

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)