A090306 a(n) = 17*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 17.

2, 17, 291, 4964, 84679, 1444507, 24641298, 420346573, 7170533039, 122319408236, 2086600473051, 35594527450103, 607193567124802, 10357885168571737, 176691241432844331, 3014108989526925364, 51416544063390575519

Offset: 0

Author

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 25 2004

Keywords

• easy
• nonn

Comments

Lim_{n-> infinity} a(n)/a(n+1) = 0.058621... = 2/(17+sqrt(293)) = (sqrt(293)-17)/2.

Lim_{n-> infinity} a(n+1)/a(n) = 17.058621... = (17+sqrt(293))/2 = 2/(sqrt(293)-17).

For more information about this type of recurrence follow the Khovanova link and see A054413, A086902 and A178765. - Johannes W. Meijer, Jun 12 2010

Examples

			a(4) = 17*a(3) + a(2) = 17*4964 + 291=((17+sqrt(293))/2)^4 + ((17-sqrt(293))/2)^4 = 84678.999988190 + 0.000011809 = 84679.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..500
• Tanya Khovanova, Recursive Sequences
• Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
• Index entries for linear recurrences with constant coefficients, signature (17,1).

Crossrefs

Cf. A005074.

Lucas polynomials Lucas(n,m): A000032 (m=1), A002203 (m=2), A006497 (m=3), A014448 (m=4), A087130 (m=5), A085447 (m=6), A086902 (m=7), A086594 (m=8), A087798 (m=9), A086927 (m=10), A001946 (m=11), A086928 (m=12), A088316 (m=13), A090300 (m=14), A090301 (m=15), A090305 (m=16), this sequence (m=17), A090307 (m=18), A090308 (m=19), A090309 (m=20), A090310 (m=21), A090313 (m=22), A090314 (m=23), A090316 (m=24), A330767 (m=25).

Programs

GAP

m:=17;; a:=[2,m];; for n in [3..20] do a[n]:=m*a[n-1]+a[n-2]; od; a; # G. C. Greubel, Dec 30 2019

Magma

m:=17; I:=[2,m]; [n le 2 select I[n] else m*Self(n-1) +Self(n-2): n in [1..20]]; // G. C. Greubel, Dec 30 2019

Maple

seq(simplify(2*(-I)^n*ChebyshevT(n, 17*I/2)), n = 0..20); # G. C. Greubel, Dec 30 2019

Mathematica

LinearRecurrence[{17,1},{2,17},30] (* Harvey P. Dale, Jan 24 2018 *)
LucasL[Range[20]-1, 17] (* G. C. Greubel, Dec 30 2019 *)

PARI

vector(21, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 17*I/2) ) \\ G. C. Greubel, Dec 30 2019

Sage

[2*(-I)^n*chebyshev_T(n, 17*I/2) for n in (0..20)] # G. C. Greubel, Dec 30 2019

Formula

a(n) = 17*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 17.

a(n) = ((17+sqrt(293))/2)^n + ((17-sqrt(293))/2)^n.

(a(n))^2 = a(2n) - 2 if n=1, 3, 5, ...

(a(n))^2 = a(2n) + 2 if n=2, 4, 6, ...

G.f.: (2-17*x)/(1-17*x-x^2). - Philippe Deléham, Nov 02 2008

From Johannes W. Meijer, Jun 12 2010: (Start)

a(2n+1) = 17*A098249(n).

a(3n+1) = A041550(5n), a(3n+2) = A041550(5n+3), a(3n+3) = 2*A041550(5n+4).

Lim_{k-> infinity} a(n+k)/a(k) = (A090306(n) + A178765(n)*sqrt(293))/2.

Lim_{n-> infinity} A090306(n)/A178765(n) = sqrt(293). (End)

a(n) = Lucas(n, 17) = 2*(-i)^n * ChebyshevT(n, 17*i/2). - G. C. Greubel, Dec 30 2019

E.g.f.: 2*exp(17*x/2)*cosh(sqrt(293)*x/2). - Stefano Spezia, Dec 31 2019

Extensions

More terms from Ray Chandler, Feb 14 2004