A289115 a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) - 2*a(n-4) + a(n-5) for n >= 10, a(0) = 2, a(1) = 4, a(2) = 7, a(3) = 12, a(4) = 20, a(5) = 32, a(6) = 50, a(7) = 77, a(8) = 116, a(9) = 174.

2, 4, 7, 12, 20, 32, 50, 77, 116, 174, 260, 386, 571, 843, 1242, 1827, 2685, 3943, 5787, 8490, 12452, 18259, 26770, 39244, 57526, 84320, 123589, 181141, 265488, 389105, 570275, 835793, 1224929, 1795236, 2631062, 3856025, 5651296, 8282394, 12138456, 17789790

Offset

0,1

Comments

Conjecture: a(n) is the number of letters (0's and 1's) in the n-th iterate of the mapping 00->0010, 01->011, 10->001, starting with 00; see A289112.

Links

Clark Kimberling, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (3, -3, 2, -2, 1).

Formula

a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) - 2*a(n-4) + a(n-5) for n >= 10, a(0) = 2, a(1) = 4, a(2) = 7, a(3) = 12, a(4) = 20, a(5) = 32, a(6) = 50, a(7) = 77, a(8) = 116, a(9) = 174.

G.f.: (2 - 2*x + x^2 - x^3 + x^4 - x^8 + x^9) / ((1 - x)^2*(1 - x - x^3)). - Colin Barker, Jul 02 2017

Mathematica

Join[{2, 4, 7, 12, 20}, LinearRecurrence[{3, -3, 2, -2, 1}, {32, 50, 77, 116, 174}, 40]]
CoefficientList[Series[(2 - 2 x + x^2 - x^3 + x^4 - x^8 + x^9) / ((1 - x)^2 (1 - x - x^3)), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 26 2017 *)

Prog

(PARI) Vec((2 - 2*x + x^2 - x^3 + x^4 - x^8 + x^9) / ((1 - x)^2*(1 - x - x^3)) + O(x^60)) \\ Colin Barker, Jul 02 2017
(Magma) I:=[2, 4, 7, 12, 20, 32, 50, 77, 116, 174]; [n le 10 select I[n] else 3*Self(n-1)-3*Self(n-2)+2*Self(n-3)-2*Self(n-4)+ Self(n-5): n in [1..40]]; // Vincenzo Librandi, Sep 26 2017

Crossrefs

Cf. A289112.

Sequence in context: A101230 A128129 A014968 * A342528 A126348 A006731

Adjacent sequences: A289112 A289113 A289114 * A289116 A289117 A289118

Keyword

nonn,easy

Author

Clark Kimberling, Jun 28 2017

Status

approved