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A090314
a(n) = 23*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 23.
13
2, 23, 531, 12236, 281959, 6497293, 149719698, 3450050347, 79500877679, 1831970236964, 42214816327851, 972772745777537, 22415987969211202, 516540496037635183, 11902847396834820411, 274282030623238504636, 6320389551731320427039, 145643241720443608326533, 3356114949121934311937298
OFFSET
0,1
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).
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
EXAMPLE
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.
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 *)
PROG
(PARI) vector(21, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 23*I/2) ) \\ 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
(Sage) [2*(-I)^n*chebyshev_T(n, 23*I/2) for n in (0..20)] # G. C. Greubel, Dec 29 2019
(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
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), 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).
Sequence in context: A167417 A053161 A090731 * A084322 A073062 A015098
KEYWORD
easy,nonn
AUTHOR
Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 25 2004
EXTENSIONS
More terms from Ray Chandler, Feb 14 2004
Terms a(16) onward added by G. C. Greubel, Dec 29 2019
STATUS
approved