A153177 a(n) = L(9*n)/L(n) where L(n) = Lucas number A000204(n).

76, 1926, 109801, 4769326, 230701876, 10716675201, 505618944676, 23714405408926, 1114769987764201, 52357935173823126, 2459933168462154076, 115560463558534156801, 5428954301161174383676, 255043991670277234750326

Offset: 1

Author

Artur Jasinski, Dec 20 2008

Keywords

• nonn

Comments

All numbers in this sequence are:

congruent to 1 mod 100 (iff n is congruent to 0 mod 3),

congruent to 26 mod 100 (iff n is congruent to 2 or 4 mod 6),

congruent to 76 mod 100 (iff n is congruent to 1 or 5 mod 6).

Links

• G. C. Greubel, Table of n, a(n) for n = 1..595
• Index entries for linear recurrences with constant coefficients, signature (34,714,-4641,-12376,12376,4641,-714,-34,1).

Crossrefs

Cf. A000032, A000204, A110391, A153173, A153175.

Cf. A153179, A153180. - R. J. Mathar, Oct 22 2010

Programs

Magma

[Lucas(9*n)/Lucas(n): n in [0..30]]; // G. C. Greubel, Dec 21 2017

Mathematica

Table[LucasL[9*n]/LucasL[n], {n, 1, 50}]
LinearRecurrence[{34,714,-4641,-12376,12376,4641,-714,-34,1},{76,1926,109801,4769326,230701876,10716675201,505618944676,23714405408926,1114769987764201},20] (* Harvey P. Dale, Aug 12 2012 *)

PARI

{lucas(n) = fibonacci(n+1) + fibonacci(n-1)};
for(n=0,30, print1( lucas(9*n)/lucas(n), ", ")) \\ G. C. Greubel, Dec 21 2017

Formula

From R. J. Mathar, Oct 22 2010: (Start)

a(n) = 34*a(n-1) +714*a(n-2) -4641*a(n-3) -12376*a(n-4) +12376*a(n-5) +4641*a(n-6) -714*a(n-7) -34*a(n-8) +a(n-9).

G.f.: -x*(76-658*x-9947*x^2+13644*x^3+26020*x^4-5306*x^5-1372*x^6+42*x^7 +x^8) / ((x-1)*(x^2+18*x+1)*(x^2-47*x+1)*(x^2+3*x+1)*(x^2-7*x+1)).

a(n) = 1-(-1)^n*A087215(n) -(-1)^n*A005248(n) +A056854(n) +A087265(n). (End)