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%I M0646 N0240 #81 Jun 28 2023 20:33:51
%S 0,2,3,5,7,10,14,20,29,43,65,100,156,246,391,625,1003,1614,2602,4200,
%T 6785,10967,17733,28680,46392,75050,121419,196445,317839,514258,
%U 832070,1346300,2178341,3524611,5702921,9227500,14930388,24157854,39088207,63246025
%N a(n) = Fibonacci(n) + n.
%C Let _x indicate the sequence offset. Then a(n+2)_0 = A006355(n+4)_0 - A066982(n+1)_1 (conjecture); (a(n)) = em[K* ]seq( .25'i - .25'j - .25'k - .25i' + .25j' - .75k' - .25'ii' - .25'jj' - .25'kk' - .25'ij' - .25'ik' - .75'ji' + .25'jk' - .25'ki' - .75'kj' + .75e), apart from initial term. - _Creighton Dement_, Nov 19 2004
%D R. Honsberger, Ingenuity in Math., Random House, 1970, p. 96.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A002062/b002062.txt">Table of n, a(n) for n = 0..500</a>
%H Hung Viet Chu, <a href="https://arxiv.org/abs/2205.14260">A Note on the Fibonacci Sequence and Schreier-type Sets</a>, arXiv:2205.14260 [math.CO], 2022.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,1).
%F G.f.: x*(-2+3*x) / ( (x^2+x-1)*(x-1)^2 ). - _Simon Plouffe_ in his 1992 dissertation
%F From _Wolfdieter Lang_: (Start)
%F Convolution of natural numbers n >= 1 with Fibonacci numbers F(k), k >= -3, (F(-k)=(-1)^(k+1)*F(k));
%F G.f.: x*(2-3*x)/((1-x-x^2)*(1-x)^2). (End)
%F a(n) = 2*a(n-1) - a(n-3) - 1. - _Kieren MacMillan_, Nov 08 2008
%F a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4). - _Emmanuel Vantieghem_, May 19 2016
%F E.g.f.: 2*exp(x/2)*sinh(sqrt(5)*x/2)/sqrt(5) + x*exp(x). - _Ilya Gutkovskiy_, Apr 11 2017
%p a:= n-> combinat[fibonacci](n)+n: seq(a(n), n=0..50); # _Zerinvary Lajos_, Mar 20 2008
%t Table[Fibonacci[n]+n,{n,0,50}] (* _Harvey P. Dale_, Jul 27 2011 *)
%o (MuPAD) numlib::fibonacci(n)+n $ n = 0..50; // _Zerinvary Lajos_, May 08 2008
%o (Haskell)
%o a002062 n = a000045 n + toInteger n
%o a002062_list = 0 : 2 : 3 : (map (subtract 1) $
%o zipWith (-) (map (* 2) $ drop 2 a002062_list) a002062_list)
%o -- _Reinhard Zumkeller_, Oct 03 2012
%o (PARI) a(n)=fibonacci(n) + n \\ _Charles R Greathouse IV_, Oct 03 2016
%o (Magma) [Fibonacci(n)+n: n in [0..50]]; // _G. C. Greubel_, Jul 09 2019
%o (Sage) [fibonacci(n)+n for n in (0..50)] # _G. C. Greubel_, Jul 09 2019
%o (GAP) List([0..50], n-> Fibonacci(n)+n) # _G. C. Greubel_, Jul 09 2019
%Y Cf. A000045, A001611, A160536, A212272.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_
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