A169986 Ceiling(phi^n) where phi = (1+sqrt(5))/2.
1, 2, 3, 5, 7, 12, 18, 30, 47, 77, 123, 200, 322, 522, 843, 1365, 2207, 3572, 5778, 9350, 15127, 24477, 39603, 64080, 103682, 167762, 271443, 439205, 710647, 1149852, 1860498, 3010350, 4870847, 7881197, 12752043, 20633240, 33385282
Offset: 0
Links
- Danny Rorabaugh, Table of n, a(n) for n = 0..4000
- Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-1).
Programs
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Magma
[1] cat [3*Fibonacci(n-1) + Fibonacci(n-2)+ n mod 2: n in [1..40]]; // Vincenzo Librandi, Apr 16 2015
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Mathematica
Ceiling[GoldenRatio^Range[0,40]] (* or *) Join[{1},LinearRecurrence[{1,2,-1,-1},{2,3,5,7},40]] (* Harvey P. Dale, Nov 12 2014 *) -
PARI
a(n)=if(n, 3*fibonacci(n-1) + fibonacci(n-2) + n%2, 1) \\ Charles R Greathouse IV, Apr 16 2015
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Sage
[ceil(golden_ratio^n) for n in range(37)] # Danny Rorabaugh, Apr 16 2015
Formula
For n >= 5, a(n) = a(n-1) + 2a(n-2) - a(n-3) - a(n-4). - Charles R Greathouse IV, Oct 14 2010
a(n) = 3*Fibonacci(n-1) + Fibonacci(n-2) + (n mod 2), n>0. - Gary Detlefs, Dec 29 2010
G.f.: (-x+x^2+x^3+x^4-1) / ((1-x)*(1+x)*(x^2+x-1)). - R. J. Mathar, Jan 06 2011
a(2k) = A000032(2k) = A169985(2k) and a(2k+1) = A000032(2k+1)+1 = A169985(2k+1)+1, for k>0. - Danny Rorabaugh, Apr 15 2015