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A020876
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a(n) = ((5+sqrt(5))/2)^n + ((5-sqrt(5))/2)^n.
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15
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2, 5, 15, 50, 175, 625, 2250, 8125, 29375, 106250, 384375, 1390625, 5031250, 18203125, 65859375, 238281250, 862109375, 3119140625, 11285156250, 40830078125, 147724609375, 534472656250, 1933740234375, 6996337890625, 25312988281250, 91583251953125
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OFFSET
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0,1
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COMMENTS
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Number of no-leaf edge-subgraphs in Moebius ladder M_n.
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LINKS
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FORMULA
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Also, a(n) = (sqrt(5)*phi)^n + (sqrt(5)/phi)^n, where phi = golden ratio. - N. J. A. Sloane, Aug 08 2014
Let S(n, m)=sum(k=0, n, binomial(n, k)*fibonacci(m*k)), then for n>0 a(n)= S(2*n, 2)/S(n, 2). - Benoit Cloitre, Oct 22 2003
a(n)= 5*a(n-1) - 5*a(n-2).
G.f.: (2-5*x)/(1-5*x+5*x^2). (End)
a((2*m+1)*k)/a(k) = Sum_{i=0..m-1} (-1)^(i*(k+1))*a(2*(m-i)*k) + 5^(m*k).
a(m+r)*a(n+s) + a(m+s)*a(n+r) = 2*a(m+n+r+s) + 5^(n+s)*a(m-n)*a(r-s).
a(n)^2 - a(n+1)*a(n-1) = -5^n.
a(n)^2 - a(n+r)*a(n-r) = -5^(n-r+1)*A093131(r)^2.
a(m)*a(n+1) - a(m+1)*a(n) = -5^(n+1)*A093131(m-n).
a(m+n) + 5^(n)*a(m-n) = a(m)*a(n).
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EXAMPLE
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G.f. = 2 + 5*x + 15*x^2 + 50*x^3 + 175*x^4 + 625*x^5 + 2250*x^6 + ...
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MAPLE
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G:=(x, n)-> cos(x)^n+cos(3*x)^n:
seq(simplify(4^n*G(Pi/10, 2*n)), n=0..22); # Gary Detlefs, Dec 05 2010
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MATHEMATICA
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Table[Sum[LucasL[2*i] Binomial[n, i], {i, 0, n}], {n, 0, 50}] (* T. D. Noe, Sep 10 2011 *)
CoefficientList[Series[(2 - 5 x)/(1 - 5 x + 5 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 08 2014 *)
LinearRecurrence[{5, -5}, {2, 5}, 30] (* Harvey P. Dale, Mar 13 2016 *)
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PROG
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(Sage) [lucas_number2(n, 5, 5) for n in range(0, 24)] # Zerinvary Lajos, Jul 08 2008
(Magma) [Floor(((5+Sqrt(5))/2)^n+((5-Sqrt(5))/2)^n): n in [0..30]]; // Vincenzo Librandi, Aug 08 2014
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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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