A167270 a(n) = Fibonacci(n + 2) + floor(n/2).

1, 2, 4, 6, 10, 15, 24, 37, 59, 93, 149, 238, 383, 616, 994, 1604, 2592, 4189, 6774, 10955, 17721, 28667, 46379, 75036, 121405, 196430, 317824, 514242, 832054, 1346283, 2178324, 3524593, 5702903, 9227481, 14930369, 24157834, 39088187, 63246004, 102334174, 165580160

Offset

0,2

Comments

a(n)/a(n-1) tends to phi = 1.6180339...; e.g., a(16)/a(15) = 2592/1604 = 1.6159...

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,1,-3,0,1).

Formula

G.f.: ( -1+x^3+x^2 ) / ( (1+x)*(x^2+x-1)*(x-1)^2 ). - R. J. Mathar, Mar 03 2013

a(n) = 2*a(n-1) + a(n-2) - 3*a(n-3) + a(n-5). - Andrew Howroyd, Aug 10 2018

Example

a(4) = 10 = (1 + 4 + 1 + 3 + 1).

Mathematica

a[n_] := Fibonacci[n+2] + Floor[n/2]; Array[a, 40, 0] (* Amiram Eldar, Jun 02 2025 *)

Prog

(PARI) a(n) = fibonacci(n+2) + n\2; \\ Andrew Howroyd, Aug 10 2018
(PARI) Vec((1 - x^2 - x^3)/((1 - x)^2*(1 + x)*(1 - x - x^2)) + O(x^40)) \\ Andrew Howroyd, Aug 10 2018

Crossrefs

Row sums of A167269.

Cf. A000045, A001622 (phi).

Sequence in context: A108925 A279026 A120549 * A355108 A060168 A113117

Adjacent sequences: A167267 A167268 A167269 * A167271 A167272 A167273

Keyword

nonn,easy

Author

Gary W. Adamson and Mats Granvik, Oct 31 2009

Extensions

Name changed and terms a(17) and beyond from Andrew Howroyd, Aug 10 2018

Status

approved