A002062 a(n) = Fibonacci(n) + n.

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Let A006355(n+4)_0%20-%20A066982(n+1)_1%20(conjecture);%20(a(n))%20=%20em%5BK*%20%5Dseq(%20.25'i%20-%20.25'j%20-%20.25'k%20-%20.25i'%20+%20.25j'%20-%20.75k'%20-%20.25'ii'%20-%20.25'jj'%20-%20.25'kk'%20-%20.25'ij'%20-%20.25'ik'%20-%20.75'ji'%20+%20.25'jk'%20-%20.25'ki'%20-%20.75'kj'%20+%20.75e),%20apart%20from%20initial%20term.%20-%20_Creighton%20Dement">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).

Crossrefs

Cf. A000045, A001611, A160536, A212272.

Programs

GAP

List([0..50], n-> Fibonacci(n)+n); # G. C. Greubel, Jul 09 2019

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

Magma

[Fibonacci(n)+n: n in [0..50]]; // G. C. Greubel, Jul 09 2019

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 *)

MuPAD

numlib::fibonacci(n)+n $ n = 0..50; // Zerinvary Lajos, May 08 2008

PARI

a(n)=fibonacci(n) + n \\ Charles R Greathouse IV, Oct 03 2016

Sage

[fibonacci(n)+n for n in (0..50)] # G. C. Greubel, Jul 09 2019

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