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A002062
a(n) = Fibonacci(n) + n.
(Formerly M0646 N0240)
13
0, 2, 3, 5, 7, 10, 14, 20, 29, 43, 65, 100, 156, 246, 391, 625, 1003, 1614, 2602, 4200, 6785, 10967, 17733, 28680, 46392, 75050, 121419, 196445, 317839, 514258, 832070, 1346300, 2178341, 3524611, 5702921, 9227500, 14930388, 24157854, 39088207, 63246025
OFFSET
0,2
COMMENTS
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
REFERENCES
R. Honsberger, Ingenuity in Math., Random House, 1970, p. 96.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Hung Viet Chu, A Note on the Fibonacci Sequence and Schreier-type Sets, arXiv:2205.14260 [math.CO], 2022.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
FORMULA
G.f.: x*(-2+3*x) / ( (x^2+x-1)*(x-1)^2 ). - Simon Plouffe in his 1992 dissertation
From Wolfdieter Lang: (Start)
Convolution of natural numbers n >= 1 with Fibonacci numbers F(k), k >= -3, (F(-k)=(-1)^(k+1)*F(k));
G.f.: x*(2-3*x)/((1-x-x^2)*(1-x)^2). (End)
a(n) = 2*a(n-1) - a(n-3) - 1. - Kieren MacMillan, Nov 08 2008
a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4). - Emmanuel Vantieghem, May 19 2016
E.g.f.: 2*exp(x/2)*sinh(sqrt(5)*x/2)/sqrt(5) + x*exp(x). - Ilya Gutkovskiy, Apr 11 2017
MAPLE
a:= n-> combinat[fibonacci](n)+n: seq(a(n), n=0..50); # Zerinvary Lajos, Mar 20 2008
MATHEMATICA
Table[Fibonacci[n]+n, {n, 0, 50}] (* Harvey P. Dale, Jul 27 2011 *)
PROG
(MuPAD) numlib::fibonacci(n)+n $ n = 0..50; // Zerinvary Lajos, May 08 2008
(Haskell)
a002062 n = a000045 n + toInteger n
a002062_list = 0 : 2 : 3 : (map (subtract 1) $
zipWith (-) (map (* 2) $ drop 2 a002062_list) a002062_list)
-- Reinhard Zumkeller, Oct 03 2012
(PARI) a(n)=fibonacci(n) + n \\ Charles R Greathouse IV, Oct 03 2016
(Magma) [Fibonacci(n)+n: n in [0..50]]; // G. C. Greubel, Jul 09 2019
(Sage) [fibonacci(n)+n for n in (0..50)] # G. C. Greubel, Jul 09 2019
(GAP) List([0..50], n-> Fibonacci(n)+n) # G. C. Greubel, Jul 09 2019
CROSSREFS
KEYWORD
nonn,easy,nice
STATUS
approved