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%I #31 Apr 23 2022 16:23:21
%S 0,1,4,34,216,1525,10336,71149,486864,3339106,22881100,156843721,
%T 1074985344,7368157369,50501844796,346145466850,2372514562656,
%U 16261461342589,111457702083424,763942486626661,5236139616899400
%N a(n) = Fibonacci(n)*Fibonacci(3n)/2.
%C This is a divisibility sequence, that is, if n | m then a(n) | a(m). However, it is not a strong divisibility sequence. It is the case k = -3 of a 1-parameter family of 4th-order linear divisibility sequences with o.g.f. x*(1 - x^2)/( (1 - k*x + x^2)*(1 - (k^2 - 2)*x + x^2) ). Compare with A000290 (case k = 2) and A215465 (case k = 3). - _Peter Bala_, Jan 17 2014
%C a(n) + a(n+1) is the numerator of the continued fraction [1,...,1,4,...,4] with n 1's followed by n 4's. - _Greg Dresden_ and _Hexuan Wang_, Aug 16 2021
%H Michael De Vlieger, <a href="/A085695/b085695.txt">Table of n, a(n) for n = 0..1197</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 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,19,4,-1).
%F G.f.: ( x - x^3 )/( 1 - 4 x - 19 x^2 - 4 x^3 + x^4 ).
%F Recurrence: a(n+4) = 4*a(n+3) + 19*a(n+2) + 4*a(n+1) - a(n).
%F a(n) = a(-n) and A153173(n) = 1 + 10*a(n) for all n in Z. - _Michael Somos_, Apr 23 2022
%e G.f. = x + 4*x^2 + 34*x^3 + 216*x^4 + 1525*x^5 + 10336*x^6 + ... - _Michael Somos_, Apr 23 2022
%t Array[Times @@ MapIndexed[Fibonacci[#]/First@ #2 &, {#, 3 #}] &, 21, 0] (* or *) LinearRecurrence[{4, 19, 4, -1}, {0, 1, 4, 34}, 21] (* or *)
%t CoefficientList[Series[(x - x^3)/(1 - 4 x - 19 x^2 - 4 x^3 + x^4), {x, 0, 20}], x] (* _Michael De Vlieger_, Dec 17 2017 *)
%o (MuPAD) numlib::fibonacci(3*n)*numlib::fibonacci(n)/2 $ n = 0..35; // _Zerinvary Lajos_, May 13 2008
%o (PARI) a(n) = fibonacci(n)*fibonacci(3*n)/2 \\ _Andrew Howroyd_, Dec 17 2017
%Y Cf. A153173, A215465.
%K easy,nonn
%O 0,3
%A _Emanuele Munarini_, Jul 18 2003
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