A090316 a(n) = 24*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 24.

2, 24, 578, 13896, 334082, 8031864, 193098818, 4642403496, 111610782722, 2683301188824, 64510839314498, 1550943444736776, 37287153512997122, 896442627756667704, 21551910219673022018, 518142287899909196136, 12456966819817493729282, 299485345963519758698904

Offset

0,1

Comments

Lim_{n->infinity} a(n)/a(n+1) = 0.0415945... = 1/(12+sqrt(145)) = sqrt(145) - 12.

Lim_{n->infinity} a(n+1)/a(n) = 24.0415945... = 12+sqrt(145) = 1/(sqrt(145)-12).

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 (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.

(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
(Sage) [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, A058168, A056949.

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).

Sequence in context: A377427 A014298 A280794 * A128578 A186632 A089835

Adjacent sequences: A090313 A090314 A090315 * A090317 A090318 A090319

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