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A090310 a(n) = 21a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 21. 2
2, 21, 443, 9324, 196247, 4130511, 86936978, 1829807049, 38512885007, 810600392196, 17061121121123, 359094143935779, 7558038143772482, 159077895163157901, 3348193836570088403, 70471148463135014364 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

a(n+1)/a(n) converges to (21+sqrt(445))/2 =21.0475115... Lim a(n)/a(n+1) as n approaches infinity = 0.0475115... = 2/(21+sqrt(445)) = (sqrt(445)-21)/2. Lim a(n+1)/a(n) as n approaches infinity = 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

LINKS

Table of n, a(n) for n=0..15.

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).

FORMULA

a(n) =21a(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). [From Philippe Deléham, Nov 02 2008]

EXAMPLE

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

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 *)

CROSSREFS

Cf. A014842, A057031.

Sequence in context: A091315 A087546 A090729 * A024232 A192666 A090451

Adjacent sequences:  A090307 A090308 A090309 * A090311 A090312 A090313

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

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Last modified April 29 17:12 EDT 2017. Contains 285607 sequences.