A085695 a(n) = Fibonacci(n)*Fibonacci(3n)/2.

0, 1, 4, 34, 216, 1525, 10336, 71149, 486864, 3339106, 22881100, 156843721, 1074985344, 7368157369, 50501844796, 346145466850, 2372514562656, 16261461342589, 111457702083424, 763942486626661, 5236139616899400

Offset: 0

Author

Emanuele Munarini, Jul 18 2003

Keywords

• easy
• nonn

Comments

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

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

Examples

			G.f. = x + 4*x^2 + 34*x^3 + 216*x^4 + 1525*x^5 + 10336*x^6 + ... - _Michael Somos_, Apr 23 2022
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..1197
• H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
• Index entries for linear recurrences with constant coefficients, signature (4,19,4,-1).

Crossrefs

Cf. A153173, A215465.

Programs

Mathematica

Array[Times @@ MapIndexed[Fibonacci[#]/First@ #2 &, {#, 3 #}] &, 21, 0] (* or *) LinearRecurrence[{4, 19, 4, -1}, {0, 1, 4, 34}, 21] (* or *)
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 *)

MuPAD

numlib::fibonacci(3*n)*numlib::fibonacci(n)/2 $ n = 0..35; // Zerinvary Lajos, May 13 2008

PARI

a(n) = fibonacci(n)*fibonacci(3*n)/2 \\ Andrew Howroyd, Dec 17 2017

Formula

G.f.: ( x - x^3 )/( 1 - 4 x - 19 x^2 - 4 x^3 + x^4 ).

Recurrence: a(n+4) = 4*a(n+3) + 19*a(n+2) + 4*a(n+1) - a(n).

a(n) = a(-n) and A153173(n) = 1 + 10*a(n) for all n in Z. - Michael Somos, Apr 23 2022