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%I #50 Sep 08 2022 08:45:09
%S 1,2,4,6,9,13,19,28,42,64,99,155,245,390,624,1002,1613,2601,4199,6784,
%T 10966,17732,28679,46391,75049,121418,196444,317838,514257,832069,
%U 1346299,2178340,3524610,5702920,9227499,14930387,24157853,39088206
%N a(n) = n + Fibonacci(n+1).
%C Row sums of triangle A135222. - _Gary W. Adamson_, Nov 23 2007
%C a(n) is the F(n+1)-th highest positive integer not equal to any a(k), 1 <= k <= n-1, where F(n) = Fibonacci numbers = A000045(n). - _Jaroslav Krizek_, Oct 28 2009
%H Vincenzo Librandi, <a href="/A081659/b081659.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,1).
%F a(n) = (sqrt(5)*(1+sqrt(5))^(n+1) - sqrt(5)*(1-sqrt(5))^(n+1))/(10*2^n) + n.
%F G.f.: (1-x-x^3)/((1-x-x^2)*(1-x)^2).
%F From _Jaroslav Krizek_, Oct 28 2009: (Start)
%F a(0) = 1, a(n) = a(n-1) + A000045(n-1) + 1 for n >= 1.
%F a(0) = 1, a(n) = a(n-1) + A000045(n+1) - A000045(n) + 1 for n >= 1.
%F a(0) = 1, a(1) = 2, a(2) = 4, a(n) = a(n-1) + a(n-2) - (n-3) n >= 3. (End)
%F E.g.f.: (1/10)*exp(-2*x/(1+sqrt(5)))*(5 - sqrt(5) + (5 + sqrt(5))*exp(sqrt(5)*x) + 10*exp((1/2)*(1+sqrt(5))*x)*x). - _Stefano Spezia_, Nov 20 2019
%p with(combinat); seq(n + fibonacci(n+1), n=0..40); # _G. C. Greubel_, Nov 20 2019
%t Table[ Fibonacci[n+1]+n, {n, 0, 38}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 03 2011 *)
%t CoefficientList[Series[(x^3+x-1)/((x-1)^2 (x^2+x-1)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Aug 10 2013 *)
%t LinearRecurrence[{3,-2,-1,1},{1,2,4,6},40] (* _Harvey P. Dale_, Mar 02 2016 *)
%o (MuPAD) numlib::fibonacci(n)+n-1 $ n = 1..48; // _Zerinvary Lajos_, May 08 2008
%o (Magma) [n+Fibonacci(n+1): n in [0..40]]; // _Vincenzo Librandi_, Aug 10 2013
%o (PARI) a(n)=n+fibonacci(n) \\ _Charles R Greathouse IV_, Oct 07 2015
%o (Sage) [n+fibonacci(n+1) for n in range(40)] # _G. C. Greubel_, Feb 12 2019
%o (GAP) List([0..40], n-> n + Fibonacci(n+1) ); # _G. C. Greubel_, Nov 20 2019
%Y Cf. A000045, A001611 (first differences), A002062, A135222.
%K nonn,easy
%O 0,2
%A _Paul Barry_, Mar 26 2003
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