A090314 a(n) = 23*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 23.

2, 23, 531, 12236, 281959, 6497293, 149719698, 3450050347, 79500877679, 1831970236964, 42214816327851, 972772745777537, 22415987969211202, 516540496037635183, 11902847396834820411, 274282030623238504636, 6320389551731320427039, 145643241720443608326533, 3356114949121934311937298

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) = 0.04339638... = 2/(23+sqrt(533)) = (sqrt(533)-23)/2.

Lim_{n -> infinity} a(n+1)/a(n) = 23.04339638... = (23+sqrt(533))/2 = 2/(sqrt(533) - 23).

Examples

			a(4) = 281959 = 23*a(3) + a(2) = 23*12236 + 531 = ((23 + sqrt(533))/2)^4 + ((23 - sqrt(533))/2)^4 = 281958.999996453 + 0.000003546 = 281959.
		

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 (23,1).

Crossrefs

Cf. A089141, A051502.

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), A090306 (m=17), A090307 (m=18), A090308 (m=19), A090309 (m=20), A090310 (m=21), A090313 (m=22), this sequence (m=23), A090316 (m=24), A330767 (m=25).

Programs

GAP

a:=[2,23];; for n in [3..20] do a[n]:=23*a[n-1]+a[n-2]; od; a; # G. C. Greubel, Dec 29 2019

Magma

I:=[2,23]; [n le 2 select I[n] else 23*Self(n-1) +Self(n-2): n in [1..20]]; // G. C. Greubel, Dec 29 2019

Maple

seq(simplify(2*(-I)^n*ChebyshevT(n, 23*I/2)), n = 0..20); # G. C. Greubel, Dec 29 2019

Mathematica

LinearRecurrence[{23,1},{2,23},20] (* Harvey P. Dale, Jul 11 2014 *)
LucasL[Range[20]-1,23] (* G. C. Greubel, Dec 29 2019 *)

PARI

vector(21, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 23*I/2) ) \\ G. C. Greubel, Dec 29 2019

Sage

[2*(-I)^n*chebyshev_T(n, 23*I/2) for n in (0..20)] # G. C. Greubel, Dec 29 2019

Formula

a(n) = 23*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 23.

a(n) = ((23 + sqrt(533))/2)^n + ((23 - sqrt(533))/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.: (2-23*x)/(1-23*x-x^2). - Philippe Deléham, Nov 02 2008

a(n) = Lucas(n, 23) = 2*(-i)^n * ChebyshevT(n, 23*i/2). - G. C. Greubel, Dec 29 2019

Extensions

More terms from Ray Chandler, Feb 14 2004

Terms a(16) onward added by G. C. Greubel, Dec 29 2019