A169622 a(n) = a(n-1) + Fibonacci(n), a(1)=5.
5, 6, 8, 11, 16, 24, 37, 58, 92, 147, 236, 380, 613, 990, 1600, 2587, 4184, 6768, 10949, 17714, 28660, 46371, 75028, 121396, 196421, 317814, 514232, 832043, 1346272, 2178312, 3524581, 5702890, 9227468, 14930355, 24157820, 39088172, 63245989, 102334158
Offset: 1
Examples
n=2: a(1)+Fibonacci(2) = 5+1 = 6. n=3: a(2)+Fibonacci(3) = 6+2 = 8.
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..250
- Index entries for linear recurrences with constant coefficients, signature (2,0,-1).
Programs
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Magma
[ n eq 1 select 5 else Self(n-1)+Fibonacci(n): n in [1..40] ]; // Klaus Brockhaus, Jan 31 2011
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Mathematica
RecurrenceTable[{a[1]==5,a[n]==a[n-1]+Fibonacci[n]},a[n],{n,40}] (* or *) LinearRecurrence[{2,0,-1},{5,6,8},40] (* Harvey P. Dale, Jul 20 2011 *) -
PARI
Vec((5 - 4*x - 4*x^2) / ((1 - x)*(1 - x - x^2)) + O(x^40)) \\ Colin Barker, Apr 20 2017
Formula
a(n) = 5 + A168193(n)/2.
a(n) = 2*a(n-1) - a(n-3) = 3 + A000045(n+2). - R. J. Mathar Dec 04 2009
G.f.: x*(-5+4*x+4*x^2) / ((1-x)*(x^2+x-1)). - R. J. Mathar Dec 04 2009
a(n) = 3 + (2^(-1-n)*((1-sqrt(5))^n*(-3+sqrt(5)) + (1+sqrt(5))^n*(3+sqrt(5)))) / sqrt(5). - Colin Barker, Apr 20 2017