A081659 a(n) = n + Fibonacci(n+1).

1, 2, 4, 6, 9, 13, 19, 28, 42, 64, 99, 155, 245, 390, 624, 1002, 1613, 2601, 4199, 6784, 10966, 17732, 28679, 46391, 75049, 121418, 196444, 317838, 514257, 832069, 1346299, 2178340, 3524610, 5702920, 9227499, 14930387, 24157853, 39088206

Offset

0,2

Comments

Row sums of triangle A135222. - Gary W. Adamson, Nov 23 2007

a(n) is the F(n+1)-th highest positive integer not equal to any a(k), 1 <= k <= n-1, where F(n) = Fibonacci numbers = A000045(n). - Jaroslav Krizek, Oct 28 2009

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

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

Formula

a(n) = (sqrt(5)*(1+sqrt(5))^(n+1) - sqrt(5)*(1-sqrt(5))^(n+1))/(10*2^n) + n.

G.f.: (1-x-x^3)/((1-x-x^2)*(1-x)^2).

From Jaroslav Krizek, Oct 28 2009: (Start)

a(0) = 1, a(n) = a(n-1) + A000045(n-1) + 1 for n >= 1.

a(0) = 1, a(n) = a(n-1) + A000045(n+1) - A000045(n) + 1 for n >= 1.

a(0) = 1, a(1) = 2, a(2) = 4, a(n) = a(n-1) + a(n-2) - (n-3) n >= 3. (End)

E.g.f.: (1/10)*exp(-2*x/(1+sqrt(5)))*(5 - sqrt(5) + (5 + sqrt(5))*exp(sqrt(5)*x) + 10*exp((1/2)*(1+sqrt(5))*x)*x). - Stefano Spezia, Nov 20 2019

Maple

with(combinat); seq(n + fibonacci(n+1), n=0..40); # G. C. Greubel, Nov 20 2019

Mathematica

Table[ Fibonacci[n+1]+n, {n, 0, 38}] (* Vladimir Joseph Stephan Orlovsky, Apr 03 2011 *)
CoefficientList[Series[(x^3+x-1)/((x-1)^2 (x^2+x-1)), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 10 2013 *)
LinearRecurrence[{3, -2, -1, 1}, {1, 2, 4, 6}, 40] (* Harvey P. Dale, Mar 02 2016 *)

Prog

(MuPAD) numlib::fibonacci(n)+n-1 $ n = 1..48; // Zerinvary Lajos, May 08 2008
(Magma) [n+Fibonacci(n+1): n in [0..40]]; // Vincenzo Librandi, Aug 10 2013
(PARI) a(n)=n+fibonacci(n) \\ Charles R Greathouse IV, Oct 07 2015
(Sage) [n+fibonacci(n+1) for n in range(40)] # G. C. Greubel, Feb 12 2019
(GAP) List([0..40], n-> n + Fibonacci(n+1) ); # G. C. Greubel, Nov 20 2019

Crossrefs

Cf. A000045, A001611 (first differences), A002062, A135222.

Sequence in context: A171861 A039900 A039902 * A143586 A363457 A344677

Adjacent sequences: A081656 A081657 A081658 * A081660 A081661 A081662

Keyword

nonn,easy

Author

Paul Barry, Mar 26 2003

Status

approved