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

%I #6 Aug 10 2018 18:10:02

%S 1,2,4,6,10,15,24,37,59,93,149,238,383,616,994,1604,2592,4189,6774,

%T 10955,17721,28667,46379,75036,121405,196430,317824,514242,832054,

%U 1346283,2178324,3524593,5702903,9227481,14930369,24157834,39088187,63246004,102334174,165580160

%N a(n) = Fibonacci(n + 2) + floor(n/2).

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

%H Andrew Howroyd, <a href="/A167270/b167270.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2, 1, -3, 0, 1)

%F G.f.: ( -1+x^3+x^2 ) / ( (1+x)*(x^2+x-1)*(x-1)^2 ). - _R. J. Mathar_, Mar 03 2013

%F a(n) = 2*a(n-1) + a(n-2) - 3*a(n-3) + a(n-5). - _Andrew Howroyd_, Aug 10 2018

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

%o (PARI) a(n) = fibonacci(n+2) + n\2; \\ _Andrew Howroyd_, Aug 10 2018

%o (PARI) Vec((1 - x^2 - x^3)/((1 - x)^2*(1 + x)*(1 - x - x^2)) + O(x^40)) \\ _Andrew Howroyd_, Aug 10 2018

%Y Row sums of A167269.

%K nonn

%O 0,2

%A _Gary W. Adamson_ & _Mats Granvik_, Oct 31 2009

%E Name changed and terms a(17) and beyond from _Andrew Howroyd_, Aug 10 2018