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A090307 a(n) = 18*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 18. 13
2, 18, 326, 5886, 106274, 1918818, 34644998, 625528782, 11294163074, 203920464114, 3681862517126, 66477445772382, 1200275886420002, 21671443401332418, 391286257110403526, 7064824071388595886 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
Lim_{n-> infinity} a(n)/a(n+1) = 0.0553851... = 1/(9+sqrt(82)) = (sqrt(82)-9).
Lim_{n-> infinity} a(n+1)/a(n) = 18.0553851... = (9+sqrt(82)) = 1/(sqrt(82)-9).
LINKS
Tanya Khovanova, Recursive Sequences
FORMULA
a(n) = 18*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 18.
a(n) = (9+sqrt(82))^n + (9-sqrt(82))^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.: (2-18*x)/(1-18*x-x^2). - Philippe Deléham, Nov 02 2008
a(n) = Lucas(n, 18) = 2*(-i)^n * ChebyshevT(n, 9*i). - G. C. Greubel, Dec 30 2019
E.g.f.: 2*exp(9*x)*cosh(sqrt(82)*x). - Stefano Spezia, Dec 31 2019
EXAMPLE
a(4) = 18*a(3) + a(2) = 18*5886 + 326 = (9+sqrt(82))^4 + (9-sqrt(82))^4 = 106273.9999905903 + 0.000009406 = 106274.
MAPLE
seq(simplify(2*(-I)^n*ChebyshevT(n, 9*I)), n = 0..20); # G. C. Greubel, Dec 30 2019
MATHEMATICA
LinearRecurrence[{18, 1}, {2, 18}, 25] (* or *) CoefficientList[ Series[ (2-18x)/(1-18x-x^2), {x, 0, 25}], x] (* Harvey P. Dale, Apr 22 2011 *)
LucasL[Range[20]-1, 18] (* G. C. Greubel, Dec 30 2019 *)
PROG
(PARI) vector(21, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 9*I) ) \\ G. C. Greubel, Dec 30 2019
(Magma) m:=18; 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
(Sage) [2*(-I)^n*chebyshev_T(n, 9*I) for n in (0..20)] # G. C. Greubel, Dec 30 2019
(GAP) m:=18;; 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
CROSSREFS
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), this sequence (m=18), A090308 (m=19), A090309 (m=20), A090310 (m=21), A090313 (m=22), A090314 (m=23), A090316 (m=24), A330767 (m=25).
Sequence in context: A192985 A193264 A191492 * A123311 A349881 A181536
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 June 14 19:19 EDT 2024. Contains 373401 sequences. (Running on oeis4.)