A090310 a(n) = 21*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 21.

2, 21, 443, 9324, 196247, 4130511, 86936978, 1829807049, 38512885007, 810600392196, 17061121121123, 359094143935779, 7558038143772482, 159077895163157901, 3348193836570088403, 70471148463135014364

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) = 0.0475115... = 2/(21+sqrt(445)) = (sqrt(445)-21)/2.

Lim_{n-> infinity} a(n+1)/a(n) = 21.0475115... = (21+sqrt(445))/2 = 2/(sqrt(445)-21).

a(2) = 443 divides a(14) = 3348193836570088403. Does this relate to the sequence being the (21,1)-weighted Fibonacci sequence with seed (2,21) and both 14 and 21 being multiples of 7? Primes in this sequence include: a(0) = 2, a(2) = 443, a(4) = 196247 Semiprimes in this sequence include: a(8) = 38512885007 = 97967 * 393121, a(14) = 3348193836570088403 = 443 * 7557999631083721. - Jonathan Vos Post, Feb 10 2005

Examples

			a(4) = 21*a(3) + a(2) = 21*9324 + 443 = ((21+sqrt(445))/2)^4 + ((21-sqrt(445))/2)^4 = 196246.9999949043 + 0.0000050956 = 196247.
		

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 (21,1).

Crossrefs

Cf. A014842, A057031.

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), this sequence (m=21), A090313 (m=22), A090314 (m=23), A090316 (m=24), A330767 (m=25).

Programs

GAP

m:=21;; 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:=21; 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, 21*I/2)), n = 0..20); # G. C. Greubel, Dec 30 2019

Mathematica

LinearRecurrence[{21,1},{2,21},40] (* or *) CoefficientList[ Series[ (2-21x)/(1-21x-x^2),{x,0,40}],x]  (* Harvey P. Dale, Apr 24 2011 *)
LucasL[Range[20]-1,21] (* G. C. Greubel, Dec 30 2019 *)

PARI

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

Sage

[2*(-I)^n*chebyshev_T(n, 21*I/2) for n in (0..20)] # G. C. Greubel, Dec 30 2019

Formula

a(n) = 21*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 21.

a(n) = ((21+sqrt(445))/2)^n + ((21-sqrt(445))/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.: (2-21*x)/(1-21*x-x^2). - Philippe Deléham, Nov 02 2008

a(n) = Lucas(n, 21) = 2*(-i)^n * ChebyshevT(n, 21*i/2). - G. C. Greubel, Dec 30 2019

Extensions

More terms from Ray Chandler, Feb 14 2004