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

Author

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 25 2004

Keywords

• easy
• nonn

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).

Examples

			a(4) =334082 = 24a(3) + a(2) = 24*13896+ 578 = (12+sqrt(145))^4 + (12-sqrt(145))^4 = 334081.99999700672 + 0.00000299327 = 334082.
		

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).

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).

Programs

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

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

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 *)

PARI

vector(21, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 12*I) ) \\ 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

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

Extensions

More terms from Ray Chandler, Feb 14 2004

Corrected by T. D. Noe, Nov 07 2006