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A239284 (15^n-(-1)^n)/16. 1
0, 1, 14, 211, 3164, 47461, 711914, 10678711, 160180664, 2402709961, 36040649414, 540609741211, 8109146118164, 121637191772461, 1824557876586914, 27368368148803711, 410525522232055664, 6157882833480834961, 92368242502212524414, 1385523637533187866211 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Let k and t be positive integers and consider a(n) = k*a(n-1)+t*a(n-2) for n>=2, with a(0)=0, a(1)=1.

The roots of its characteristic equation are r1 = (k+sqrt(k^2+4t))/2 and r2 =(k-sqrt(k^2+4t))/2. Hence, the solution to the recurrence relation is the sequence {a(n)} where a(n) = alpha1*r1^n + alpha2*r2^n. It can be shown that alpha1 = 1/sqrt(k^2+4t) and alpha2 = -alpha1. It can be shown also that |r2/r1|< 1. Thus, the ratio a(n+1)/a(n) converges to r as n approaches infinity.

Note that limit a(n+1)/a(n) = 15 as n approaches infinity with k=14 and t=15.

If n > 15 then | a(n+1)/a(n) - 15 | < 10^(-16).

LINKS

Table of n, a(n) for n=0..19.

FORMULA

G.f.: x/(1 -14*x - 15*x^2).

a(n) = 14*a(n-1) + 15*a(n-2) for n > 1, a(0)=0, a(1)=1.

a(n) = (1/16)*(15^n - (-1)^n).

a(n) = (1/16)*( A001024(n) - A033999(n) ).

PROG

(PARI) a(n) = (15^n - (-1)^n)/16; \\ Michel Marcus, Mar 16 2014

CROSSREFS

Sequence in context: A158555 A097183 A004369 * A241260 A240326 A202976

Adjacent sequences:  A239281 A239282 A239283 * A239285 A239286 A239287

KEYWORD

nonn,easy

AUTHOR

Felix P. Muga II, Mar 14 2014

STATUS

approved

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Last modified February 21 22:56 EST 2018. Contains 299427 sequences. (Running on oeis4.)