A002062 a(n) = Fibonacci(n) + n. (Formerly M0646 N0240)

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

T. D. Noe, Table of n, a(n) for n = 0..500

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

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

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

Cf. A000045, A001611, A160536, A212272.

Sequence in context: A214077 A094984 A107332 * A005688 A241550 A319564

Adjacent sequences: A002059 A002060 A002061 * A002063 A002064 A002065

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved