A081662 Partial sums of n + Fibonacci(n+1).
1, 3, 7, 13, 22, 35, 54, 82, 124, 188, 287, 442, 687, 1077, 1701, 2703, 4316, 6917, 11116, 17900, 28866, 46598, 75277, 121668, 196717, 318135, 514579, 832417, 1346674, 2178743, 3525042, 5703382, 9227992, 14930912, 24158411, 39088798
Offset: 0
Links
- Harvey P. Dale, Table of n, a(n) for n = 0..1000
- W. Kuszmaul, Fast Algorithms for Finding Pattern Avoiders and Counting Pattern Occurrences in Permutations, arXiv preprint arXiv:1509.08216 [cs.DM], 2015-2017.
- Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).
Programs
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Mathematica
Accumulate[Table[Total[{n,Fibonacci[n+1]}],{n,0,40}]] (* or *) LinearRecurrence[ {4,-5,1,2,-1},{1,3,7,13,22},41] (* Harvey P. Dale, Nov 19 2011 *)
Formula
a(n) = (1 - 2*sqrt(5)/5)*(sqrt(5)/2 - 1/2)^n*(-1)^n + (sqrt(5)/2 + 1/2)^n*(2*sqrt(5)/5 + 1) + (n^2 + n - 2)/2.
G.f.: (x^3 + x - 1)/((1-x)^3*(x^2 + x - 1)).
a(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 2*a(n-4) - a(n-5); a(0)=1, a(1)=3, a(2)=7, a(3)=13, a(4)=22. - Harvey P. Dale, Nov 19 2011
E.g.f.: exp(x)*(x^2 + 2*x - 2)/2 + 2*exp(x/2)*(5*cosh(sqrt(5)*x/2) + 2*sqrt(5)*sinh(sqrt(5)*x/2))/5. - Stefano Spezia, Feb 13 2023