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A002062 a(n) = n-th Fibonacci number + n.
(Formerly M0646 N0240)
12
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 (list; graph; refs; listen; history; text; internal format)
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

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

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

FORMULA

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). - Wolfdieter Lang

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

MAPLE

A002062:=z*(-2+3*z)/(z**2+z-1)/(z-1)**2; # conjectured by Simon Plouffe in his 1992 dissertation

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]+a[n-2] od: seq(a[n]+n, n=0..29); # Zerinvary Lajos, Mar 20 2008

MATHEMATICA

lst={}; Do[f=Fibonacci[n]+n; AppendTo[lst, f], {n, 0, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Mar 20 2009 *)

Table[Fibonacci[n]+n, {n, 0, 30}] (* Harvey P. Dale, Jul 27 2011 *)

PROG

(Mupad) numlib::fibonacci(n)+n $ n = 0..48; // 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

CROSSREFS

Cf. A001611, A160536, A212272.

Sequence in context: A214077 A094984 A107332 * A005688 A241550 A221943

Adjacent sequences:  A002059 A002060 A002061 * A002063 A002064 A002065

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified March 27 08:56 EDT 2017. Contains 284146 sequences.