A128533 a(n) = F(n)*L(n+2) where F=Fibonacci and L=Lucas numbers.
0, 4, 7, 22, 54, 145, 376, 988, 2583, 6766, 17710, 46369, 121392, 317812, 832039, 2178310, 5702886, 14930353, 39088168, 102334156, 267914295, 701408734, 1836311902, 4807526977, 12586269024, 32951280100, 86267571271, 225851433718, 591286729878
Offset: 0
Examples
a(4) = 54 because F(4)*L(6) = 3*18. G.f. = 4*x + 7*x^2 + 22*x^3 + 54*x^4 + 145*x^5 + 376*x^6 + 988*x^7 + ...
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Mohammad K. Azarian, Identities Involving Lucas or Fibonacci and Lucas Numbers as Binomial Sums, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 45, 2012, pp. 2221-2227.
- Index entries for linear recurrences with constant coefficients, signature (2,2,-1).
Programs
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GAP
List([0..30], n -> Fibonacci(2*(n+1)) + (-1)^(n+1)); # G. C. Greubel, Jan 07 2019
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Magma
[Fibonacci(n)*Lucas(n+2): n in [0..30]]; // Vincenzo Librandi, Feb 20 2013
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Maple
with(combinat); A128533:=n->fibonacci(2*n+2)+(-1)^(n+1); seq(A128533(k),k=0..50); # Wesley Ivan Hurt, Oct 19 2013
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Mathematica
LinearRecurrence[{2,2,-1}, {0,4,7}, 40] (* Vincenzo Librandi, Feb 20 2013 *) a[n_]:= Fibonacci[2n+2] -(-1)^n; (* Michael Somos, May 26 2014 *) -
PARI
vector(30, n, n--; fibonacci(2*(n+1)) + (-1)^(n+1)) \\ G. C. Greubel, Jan 07 2019
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Sage
[fibonacci(2*(n+1)) + (-1)^(n+1) for n in (0..30)] # G. C. Greubel, Jan 07 2019
Formula
a(n) = F(2*(n+1)) + (-1)^(n+1), assuming F(0) = 0 and L(0) = 2.
From R. J. Mathar, Apr 16 2009: (Start)
a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
G.f.: x*(4-x)/((1+x)*(x^2-3*x+1)). (End)
a(n) = A186679(2*n). - Reinhard Zumkeller, Feb 25 2011
a(-n) = - A128535(n). - Michael Somos, May 26 2014
0 = a(n)*(+4*a(n) + a(n+1) - 17*a(n+2)) + a(n+1)*(-14*a(n+1) + a(n+2)) + a(n+2)*(+4*a(n+2)) for all n in Z. - Michael Somos, May 26 2014
Extensions
More terms from Vincenzo Librandi, Feb 20 2013
Comments