A319172 a(n) = 2*(a(n-1) + a(n-3)) - a(n-4), with a(0) = 1, a(1) = 2, a(2) = 5 and a(3) = 12.

1, 2, 5, 12, 27, 62, 143, 328, 753, 1730, 3973, 9124, 20955, 48126, 110527, 253840, 582977, 1338882, 3074917, 7061948, 16218683, 37248318, 85545615, 196466648, 451211249, 1036265410, 2379918501, 5465792852, 12552905275, 28829382142

Offset

0,2

Comments

The growth factor for large n (i.e., a(n)/a(n-1)) is (1 + sqrt(3) + sqrt(2*sqrt(3))) = A319129.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,0,2,-1).

Formula

Limit_{n -> inf} a(n)/a(n-1) = (1 + sqrt(3) + sqrt(2*sqrt(3)))/2 = A319129.

G.f.: (1 + x^2) / (1 - 2*x - 2*x^3 + x^4). - Colin Barker, Sep 13 2018

Maple

f:= gfun:-rectoproc({a(n) = 2*(a(n-1)+a(n-3))-a(n-4), a(0) = 1, a(1) = 2, a(2) = 5, a(3) = 12}, a(n), remember):
map(f, [$0..40]); # Robert Israel, Sep 12 2018

Mathematica

LinearRecurrence[{2, 0, 2, -1}, {1, 2, 5, 12}, 30] (* Vincenzo Librandi, Sep 29 2018 *)
CoefficientList[Series[(1 + x^2) / (1 - 2*x - 2*x^3 + x^4), {x, 0, 30}], x] (* Stefano Spezia, Sep 29 2018 *)

Prog

(GAP) a:=[1, 2, 5, 12];; for n in [5..30] do a[n]:=2*(a[n-1]+a[n-3])-a[n-4]; od; a; # Muniru A Asiru, Sep 12 2018
(PARI) Vec((1 + x^2) / (1 - 2*x - 2*x^3 + x^4) + O(x^40)) \\ Colin Barker, Sep 13 2018
(Magma) I:=[1, 2, 5, 12]; [n le 4 select I[n] else 2*(Self(n-1) + Self(n-3)) - Self(n-4): n in [1..30]]; // Vincenzo Librandi, Sep 29 2018

Crossrefs

Cf. A319129.

Sequence in context: A190171 A091596 A077863 * A018009 A128096 A018010

Adjacent sequences: A319169 A319170 A319171 * A319173 A319174 A319175

Keyword

nonn,easy,changed

Author

A.H.M. Smeets, Sep 12 2018

Status

approved