A049678 - OEIS

%I #28 Sep 08 2022 08:44:58

%S 1,48,2255,105937,4976784,233802911,10983760033,516002918640,

%T 24241153416047,1138818207635569,53500214605455696,

%U 2513371268248782143,118074949393087305025,5547009250206854554032,260591359810329076734479,12242246901835259751966481

%N a(n) = F(8*n+4)/3, where F=A000045 (the Fibonacci sequence).

%H Indranil Ghosh, <a href="/A049678/b049678.txt">Table of n, a(n) for n = 0..596</a>

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H H. C. Williams and R. K. Guy, <a href="http://dx.doi.org/10.1142/S1793042111004587">Some fourth-order linear divisibility sequences</a>, Intl. J. Number Theory 7 (5) (2011) 1255-1277.

%H H. C. Williams and R. K. Guy, <a href="http://www.emis.de/journals/INTEGERS/papers/a17self/a17self.Abstract.html ">Some Monoapparitic Fourth Order Linear Divisibility Sequences</a> Integers, Volume 12A (2012) The John Selfridge Memorial Volume

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (47,-1).

%F a(n) = 47*a(n-1) - a(n-2), n>1. a(0)=1, a(1)=48.

%F G.f.: (1+x)/(1-47*x+x^2).

%F From _Peter Bala_, Mar 23 2015: (Start)

%F a(n) = A004187(2*n + 1); a(n) = A099483(4*n + 1).

%F a(n) = ( Fibonacci(8*n + 8 - 2*k) + Fibonacci(8*n + 2*k) )/( Fibonacci(8 - 2*k) + Fibonacci(2*k) ), for k an arbitrary integer.

%F a(n) = ( Fibonacci(8*n + 8 - 2*k - 1) - Fibonacci(8*n + 2*k + 1) )/( Fibonacci(8 - 2*k - 1) - Fibonacci(2*k + 1) ), for k an arbitrary integer.

%F The aerated sequence (b(n))n>=1 = [1, 0, 48, 0, 2255, 0, 105937, 0, ...] is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). It is the case P1 = 0, P2 = -45, Q = -1 of the 3-parameter family of divisibility sequences found by Williams and Guy. See A100047 for the connection with Chebyshev polynomials. (End)

%e a(2) = F(8 * 2 + 4) / 3 = F(20) / 3 = 6765 / 3 = 2255. - _Indranil Ghosh_, Feb 04 2017

%t CoefficientList[Series[(1+x)/(1-47x+x^2),{x,0,20}],x]  (* _Harvey P. Dale_, Feb 18 2011 *)

%t Table[Fibonacci[8*n+4]/3, {n,0,30}] (* _G. C. Greubel_, Dec 02 2017 *)

%o (PARI) for(n=0,30, print1(fibonacci(8*n+4)/3, ", ")) \\ _G. C. Greubel_, Dec 02 2017

%o (Magma) [Fibonacci(8*n+4)/3: n in [0..30]]; // _G. C. Greubel_, Dec 02 2017

%Y Cf. A004187, A099483, A100047.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_

%E Better description and more terms from _Michael Somos_

%E 2 more terms from _Indranil Ghosh_, Feb 04 2017