A090301 - OEIS

A090301

a(n) = 15*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 15.

2, 15, 227, 3420, 51527, 776325, 11696402, 176222355, 2655031727, 40001698260, 602680505627, 9080209282665, 136805819745602, 2061167505466695, 31054318401746027, 467875943531657100, 7049193471376602527

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OFFSET

0,1

COMMENTS

Lim_{n-> infinity} a(n)/a(n+1) = 0.066372... = 2/(15+sqrt(229)) = (sqrt(229)-15)/2.

Lim_{n-> infinity} a(n+1)/a(n) = 15.066372... = (15+sqrt(229))/2 = 2/(sqrt(229)-15).

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

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 (15,1).

FORMULA

a(n) = 15*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 15.

a(n) = ((15+sqrt(229))/2)^n + ((15-sqrt(229))/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.: (2-15*x)/(1-15*x-x^2). - Philippe Deléham, Nov 02 2008

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

Lim_{k-> infinity} a(n+k)/a(k) = (A090301(n) + A154597(n)*sqrt(229))/2.

Lim_{n-> infinity} A090301(n)/ A154597(n) = sqrt(229).

a(2n+1) = 15*A098246(n).

a(3n+1) = A041426(5n), a(3n+2) = A041426(5n+3), a(3n+3) = 2*A041426(5n+4).

(End)

a(n) = Lucas(n, 15) = 2*(-i)^n * ChebyshevT(n, 15*i/2). - G. C. Greubel, Dec 31 2019

E.g.f.: 2*exp(15*x/2)*cosh(sqrt(229)*x/2). - Stefano Spezia, Jan 01 2020

EXAMPLE

a(4) = 15*a(3) + a(2) = 15*3420 + 227 = ((15+sqrt(229))/2)^4 + ((15-sqrt(229))/2)^4 = 51526.9999805 + 0.0000194 = 51527.

MAPLE

seq(simplify(2*(-I)^n*ChebyshevT(n, 15*I/2)), n = 0..20); # G. C. Greubel, Dec 31 2019

MATHEMATICA

LucasL[Range[20]-1, 15] (* G. C. Greubel, Dec 31 2019 *)

PROG

(PARI) vector(21, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 15*I/2) ) \\ G. C. Greubel, Dec 3012019

(Magma) m:=15; 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 31 2019

(Sage) [2*(-I)^n*chebyshev_T(n, 15*I/2) for n in (0..20)] # G. C. Greubel, Dec 31 2019

(GAP) m:=15;; a:=[2, m];; for n in [3..20] do a[n]:=m*a[n-1]+a[n-2]; od; a; # G. C. Greubel, Dec 31 2019

CROSSREFS

Cf. A058087, A071416.

Lucas polynomials: A114525.

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), this sequence (m=15), A090305 (m=16), A090306 (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), A087281 (m=29), A087287 (m=76), A089772 (m=199).

Sequence in context: A176337 A145168 A184357 * A297087 A247660 A197236

Adjacent sequences: A090298 A090299 A090300 * A090302 A090303 A090304

KEYWORD

easy,nonn

AUTHOR

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

EXTENSIONS

More terms from Ray Chandler, Feb 14 2004

STATUS

approved