A090313 a(n) = 22*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 22.

2, 22, 486, 10714, 236194, 5206982, 114789798, 2530582538, 55787605634, 1229857906486, 27112661548326, 597708411969658, 13176697724880802, 290485058359347302, 6403847981630521446, 141175140654230819114

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) = 0.045361... = 1/(11+sqrt(122)) = (sqrt(122)-11).

Lim_{n-> infinity} a(n+1)/a(n) = 22.045361... = (11+sqrt(122)) = 1/(sqrt(122)-11).

Examples

			a(4) = 236194 = 22*a(3) + a(2) = 22*10714 + 486 = (11 + sqrt(122))^4 + (11 - sqrt(122))^4 = 236193.999995766 + 0.000004233 = 236194.
		

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 (22,1).

Crossrefs

Cf. A079219.

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), this sequence (m=22), A090314 (m=23), A090316 (m=24), A330767 (m=25).

Programs

GAP

m:=22;; 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:=22; 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, 11*I)), n = 0..20); # G. C. Greubel, Dec 30 2019

Mathematica

LucasL[Range[20]-1,22] (* G. C. Greubel, Dec 29 2019 *)

PARI

vector(21, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 11*I) ) \\ G. C. Greubel, Dec 30 2019

Sage

[2*(-I)^n*chebyshev_T(n, 11*I) for n in (0..20)] # G. C. Greubel, Dec 30 2019

Formula

a(n) = 22*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 22.

a(n) = (11+sqrt(122))^n + (11-sqrt(122))^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.: (2-22*x)/(1-22*x-x^2). - Philippe Deléham, Nov 02 2008

a(n) = Lucas(n, 22) = 2*(-i)^n * ChebyshevT(n, 11*i). - G. C. Greubel, Dec 30 2019

Extensions

More terms from Ray Chandler, Feb 14 2004