OFFSET
0,1
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..500
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (24,1).
FORMULA
a(n) = 24*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 24.
a(n) = (12+sqrt(145))^n + (12-sqrt(145))^n.
Limit_{n->infinity} a(n)/a(n+1) = 0.0415945... = 1/(12+sqrt(145)) = sqrt(145) - 12.
Limit_{n->infinity} a(n+1)/a(n) = 24.0415945... = 12+sqrt(145) = 1/(sqrt(145)-12).
(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*(1-12*x)/(1-24*x-x^2). - Philippe Deléham, Nov 02 2008
a(n) = 2*(-i)^n * ChebyshevT(n, 12*i) = Lucas(n, 24). - G. C. Greubel, Dec 29 2019
a(n) = 2 * A041264(n-1) for n>0. - Alois P. Heinz, Dec 29 2019
EXAMPLE
a(4) =334082 = 24a(3) + a(2) = 24*13896+ 578 = (12+sqrt(145))^4 + (12-sqrt(145))^4 = 334081.99999700672 + 0.00000299327 = 334082.
MAPLE
seq(simplify(2*(-I)^n*ChebyshevT(n, 12*I)), n = 0..20); # G. C. Greubel, Dec 29 2019
MATHEMATICA
LinearRecurrence[{24, 1}, {2, 24}, 20] (* Harvey P. Dale, Aug 30 2015 *)
LucasL[Range[20]-1, 24] (* G. C. Greubel, Dec 29 2019 *)
PROG
(PARI) vector(21, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 12*I) ) \\ G. C. Greubel, Dec 29 2019
(Magma) I:=[2, 24]; [n le 2 select I[n] else 24*Self(n-1) +Self(n-2): n in [1..20]]; // G. C. Greubel, Dec 29 2019
(SageMath) [2*(-I)^n*chebyshev_T(n, 12*I) for n in (0..20)] # G. C. Greubel, Dec 29 2019
(GAP) a:=[2, 24];; for n in [3..20] do a[n]:=24*a[n-1]+a[n-2]; od; a; # G. C. Greubel, Dec 29 2019
CROSSREFS
Cf. A041264.
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), A090314 (m=23), this sequence (m=24), A330767 (m=25).
KEYWORD
easy,nonn
AUTHOR
Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 25 2004
EXTENSIONS
More terms from Ray Chandler, Feb 14 2004
Corrected by T. D. Noe, Nov 07 2006
STATUS
approved