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A100545
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Expansion of (7-2*x) / (1-3*x+x^2).
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7
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7, 19, 50, 131, 343, 898, 2351, 6155, 16114, 42187, 110447, 289154, 757015, 1981891, 5188658, 13584083, 35563591, 93106690, 243756479, 638162747, 1670731762, 4374032539, 11451365855, 29980065026, 78488829223, 205486422643, 537970438706, 1408424893475, 3687304241719, 9653487831682, 25273159253327
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OFFSET
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0,1
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COMMENTS
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A Floretion integer sequence relating to Fibonacci numbers.
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LINKS
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FORMULA
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a(n-1) = 4*Fibonacci(2*n) + Fibonacci(2*n-1) + Fibonacci(2*n+1).
a(n) = 3*a(n-1) - a(n-2) with a(0)=7 and a(1)=19. - Philippe Deléham, Nov 16 2008
a(n) = (2^(-1-n)*((3-sqrt(5))^n*(-17+7*sqrt(5)) + (3+sqrt(5))^n*(17+7*sqrt(5)))) / sqrt(5). - Colin Barker, Oct 14 2015
a(n) = Fibonacci(2*n+4) + Lucas(2*n+3).
E.g.f.: 2*exp(3*t/2)*(cosh(sqrt(5)*t/2) + (4/sqrt(5))*sinh(sqrt(5)*t/2)). (End)
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MAPLE
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F := proc(n) combinat[fibonacci](n) ; end: A100545 := proc(n) 4*F(2*(n+1)) + F(2*n+1)+F(2*n+3) ; end: for n from 0 to 30 do printf("%d, ", A100545(n)) ; od ; # R. J. Mathar, Oct 26 2006
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MATHEMATICA
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Table[Fibonacci[2*(n+2)] + LucasL[2*n+3], {n, 0, 30}] (* G. C. Greubel, Jan 17 2020 *)
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PROG
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(PARI) Vec((7-2*x)/(1-3*x+x^2) + O(x^30)) \\ Michel Marcus, Feb 11 2015
(Magma) [Fibonacci(2*n+4) +Lucas(2*n+3): n in [0..30]]; // G. C. Greubel, Jan 17 2020
(Sage) [fibonacci(2*n+4) +lucas_number2(2*n+3, 1, -1) for n in (0..30)] # G. C. Greubel, Jan 17 2020
(GAP) List([0..30], n-> Fibonacci(2*n+4) +Lucas(1, -1, 2*n+3)[2] ); # G. C. Greubel, Jan 17 2020
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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