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
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).
Programs
-
GAP
List([0..40], n-> n + Fibonacci(n+1) ); # G. C. Greubel, Nov 20 2019
-
Magma
[n+Fibonacci(n+1): n in [0..40]]; // Vincenzo Librandi, Aug 10 2013
-
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 *) -
MuPAD
numlib::fibonacci(n)+n-1 $ n = 1..48; // Zerinvary Lajos, May 08 2008
-
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
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(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
Comments